From free.leekai@gmail.com Wed Feb  8 21:10:50 2006
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From: lee-kai <randomly@mit.edu>
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Subject: ln2 comments for reading
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I think there should be a period at the end of "you can solve at least one
problem we come up with" on page 5.

Maybe also after "$B"O(Bx $B":(B D. P (n)" and "$B"P(Bx $B":(B D. P (n)".  There are probably
some other places that could use them (like the bottom of page 6), but
finding places for periods is boring.

I found the section about the consistency of ZFC axioms and about the
Banach-Tarski Theorem the most interesting!

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I think there should be a period at the end of &quot;you can solve at least one problem we come up with&quot; on page 5.<br>
<br>
Maybe also after &quot;$B"O(Bx $B":(B D. P (n)&quot; and &quot;$B"P(Bx $B":(B D. P (n)&quot;.&nbsp; There are
probably some other places that could use them (like the bottom of page
6), but finding places for periods is boring.<br>
<br>
I found the section about the consistency of ZFC axioms and about the Banach-Tarski Theorem the most interesting!<br>

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From rfh08@MIT.EDU Thu Feb  9 23:06:51 2006
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The most surprising part of the reading was that proving 2 + 2 = 4 
requires more than 20,000 steps. That's a lot more than I would have 
expected (I would have thought a lot more than just, say, 2 or 3 steps, 
but far less than 20,000). I'm also curious as to how the figure of 
20,000 was reached.

From dbw@MIT.EDU Fri Feb 10 10:36:07 2006
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Date: Fri, 10 Feb 2006 10:36:03 -0500
From: Daniel B Wilson <dbw@MIT.EDU>
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Subject: Week 1 reading
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Professor Meyer,

I found the reading interesting. I do have a question about an example you used
regarding Iff statements. The second example used states:

"Two triangles have the same side lengths if and only if all angles are the
same."

While I agree this is a true statement, I am not sure that the reverse is always
true. Two triangles with the same angles are similar, but not necessarily
congruent.

My first question is, should the statement read something like "Two triangles
have the same side lengths if and only if all angles are the same and one side
is equal." or something thereabouts? Or is it fine the way it is.

My second question is if a statement is not necessarily correct in the reverse,
is it an Iff?

Otherwise, I found the reading rather informative and appreciated the use or
common language when able (as is necessary for a good proof).

-Daniel

From meyer@csail.mit.edu Sat Feb 11 15:57:35 2006
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Thanks for pointing out the several you found.  Good work -- and keep 
it up!

Regards, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307


From meyer@csail.mit.edu Sat Feb 11 16:12:49 2006
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On Feb 10, 2006, at 10:36 AM, Daniel B Wilson wrote:

> Professor Meyer,
>
> I found the reading interesting. I do have a question about an example 
> you used
> regarding Iff statements. The second example used states:
>
> "Two triangles have the same side lengths if and only if all angles 
> are the
> same."
>
> While I agree this is a true statement, I am not sure that the reverse 
> is always
> true. Two triangles with the same angles are similar, but not 
> necessarily
> congruent.
YES, same angles impiles similar, not congruent.  BUT that means the 
"iff" is FALSE: it's only an IMPLIES.
I've fixed it in the notes so it is an IFF along the lines you 
suggested.
>
> My first question is, should the statement read something like "Two 
> triangles
> have the same side lengths if and only if all angles are the same and 
> one side
> is equal." or something thereabouts? Or is it fine the way it is.
>
> My second question is if a statement is not necessarily correct in the 
> reverse,
> is it an Iff?
IFF means true in both directions.

> Otherwise, I found the reading rather informative and appreciated the 
> use or
> common language when able (as is necessary for a good proof).
>
> -Daniel


From meyer@csail.mit.edu Sat Feb 11 16:15:38 2006
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On Feb 9, 2006, at 11:06 PM, Richard Hermosillo wrote:

> The most surprising part of the reading was that proving 2 + 2 = 4 
> requires more than 20,000 steps. That's a lot more than I would have 
> expected (I would have thought a lot more than just, say, 2 or 3 
> steps, but far less than 20,000). I'm also curious as to how the 
> figure of 20,000 was reached.

I'm not sure I believe the 20,000 figure myself, but I have been told 
reliably that the 2+2=4 proof does not appear until several hundred 
pages into Principia Mathematica.

regards, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307

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On Feb 9, 2006, at 11:06 PM, Richard Hermosillo wrote:


<excerpt>The most surprising part of the reading was that proving 2 +
2 = 4 requires more than 20,000 steps. That's a lot more than I would
have expected (I would have thought a lot more than just, say, 2 or 3
steps, but far less than 20,000). I'm also curious as to how the
figure of 20,000 was reached.

</excerpt>

I'm not sure I believe the 20,000 figure myself, but I have been told
reliably that the 2+2=4 proof does not appear until several hundred
pages into <italic>Principia Mathematica.

</italic>

regards, A.


Prof. Albert R. Meyer

MIT Computer Science & AI Laboratory

The Stata Center, 32-G624

32 Vassar Street

Cambridge, MA 02139-4307

--Boundary_(ID_G1u/cm13eNd7qBDjtlc9aw)--

From meyer@csail.mit.edu Sat Feb 11 16:23:35 2006
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On Feb 8, 2006, at 9:10 PM, lee-kai wrote:

> I think there should be a period at the end of "you can solve at least 
> one problem we come up with" on page 5.
yes; added.
>
>  Maybe also after "$B"O(Bx $B":(B D. P (n)" and "$B"P(Bx $B":(B D. P (n)".  There are 
> probably some other places that could use them (like the bottom of 
> page 6), but finding places for periods is boring.
maybe, but here I prefer not to have the final periods.

>
>  I found the section about the consistency of ZFC axioms and about the 
> Banach-Tarski Theorem the most interesting!

Thanks, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307


From mongoose@MIT.EDU Sun Feb 12 15:50:06 2006
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-p. 8 - proof hard to follow
-p. 11 - don't understand last line
-not clear on distinction between a function f and function f with a hat

From mdburro@MIT.EDU Sun Feb 12 16:04:52 2006
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Subject: Reading assignment #1
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First of all I would like to state that not enough was said about "implies" and
-> which became exceptionally frustrating on the online problem set.  I had
thought that the final part on the first problem required a -> since "getting
an A in the class" was dependent on "getting an A on the final" and "doing all
the questions".

In the reading itself, it was not clear what would we should be doing with
proofs.  Moreover, when we should be using proof by contradiction and when we
should use direct proof.  Obviously direct proof is desired more highly than
PBC, but when should we not try to use direct proof?  What should we be looking
for when trying to do a proof, ie what should we be looking for to know to do it
with direct proof or PBC?  I think a little more clarification on that would be
helpful.

Also, the functions made absolutely no sense.  I have no idea what ::= means and
the reading came no where close to clarifying the matter.  there is no
explanation of this new and ill-defined symbol.  What does it mean?

Mark Burroughs

From meyer@csail.mit.edu Mon Feb 13 00:45:55 2006
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Begin forwarded message:

> From: Albert R. Meyer <meyer@csail.mit.edu>
> Date: February 13, 2006 12:08:45 AM EST
> To: Jorge L De la Garza <mongoose@MIT.EDU>
> Subject: Re: Comments on week 2 notes
>
>
> On Feb 12, 2006, at 3:50 PM, Jorge L De la Garza wrote:
>
>> -p. 8 - proof hard to follow
> yeah, it is pretty abstract and full of symbols
>> -p. 11 - don't understand last line
> It means that to compute f4(T,F), for example, you find the value of 
> [T-->F], which i s F by the rules for implication, and if yuo look at 
> the table, f(T,F) is indeed F.  Does that help?
>
>> -not clear on distinction between a function f and function f with a 
>> hat
> If it remains unclear after rereading. I encourage you to take it up 
> with your TA or me in office hours.
>
> regards, A.
>
> Prof. Albert R. Meyer
> MIT Computer Science & AI Laboratory
> The Stata Center, 32-G624
> 32 Vassar Street
> Cambridge, MA 02139-4307
>

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Begin forwarded message:


<excerpt><bold><color><param>0000,0000,0000</param>From:
</color></bold>Albert R. Meyer <<meyer@csail.mit.edu>

<bold><color><param>0000,0000,0000</param>Date:
</color></bold>February 13, 2006 12:08:45 AM EST

<bold><color><param>0000,0000,0000</param>To: </color></bold>Jorge L
De la Garza <<mongoose@MIT.EDU>

<bold><color><param>0000,0000,0000</param>Subject: </color>Re:
Comments on week 2 notes

</bold>


On Feb 12, 2006, at 3:50 PM, Jorge L De la Garza wrote:


<excerpt>-p. 8 - proof hard to follow

</excerpt>yeah, it is pretty abstract and full of symbols

<excerpt>-p. 11 - don't understand last line

</excerpt>It means that to compute f4(T,F), for example, you find the
value of [T-->F], which i s F by the rules for implication, and if yuo
look at the table, f(T,F) is indeed F.  Does that help?


<excerpt>-not clear on distinction between a function f and function f
with a hat

</excerpt>If it remains unclear after rereading. I encourage you to
take it up with your TA or me in office hours.


regards, A.


Prof. Albert R. Meyer

MIT Computer Science & AI Laboratory

The Stata Center, 32-G624

32 Vassar Street

Cambridge, MA 02139-4307


</excerpt>

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From meyer@imap.theory.csail.mit.edu  Mon Feb 13 00:44:41 2006
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Re: Reading assignment #1
Date: Mon, 13 Feb 2006 00:44:41 -0500
To: Mark D Burroughs <mdburro@MIT.EDU>
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On Feb 12, 2006, at 4:04 PM, Mark D Burroughs wrote:

> First of all I would like to state that not enough was said about 
> "implies" and
> -> which became exceptionally frustrating on the online problem set.  
> I had
> thought that the final part on the first problem required a -> since 
> "getting
> an A in the class" was dependent on "getting an A on the final" and 
> "doing all
> the questions".
Don't let the online probs frustrate you.  As explained in courseinfo, 
they are graded based only on submission, not correctness.  The point 
of the online probs is to call attention to important concepts in the 
notes.  You've correctly identified the behavior of "implies" as one of 
those concepts.
>
> In the reading itself, it was not clear what would we should be doing 
> with
> proofs.  Moreover, when we should be using proof by contradiction and 
> when we
> should use direct proof.  Obviously direct proof is desired more 
> highly than
> PBC, but when should we not try to use direct proof?  What should we 
> be looking
> for when trying to do a proof, ie what should we be looking for to 
> know to do it
> with direct proof or PBC?  I think a little more clarification on that 
> would be
> helpful.
Beginning students often want tp reduce proving to mechanical 
application of recipes and templates, but I don't think that works very 
well.  Makes me wonder whether you would ask for a recipe for when a 
programmer should use a FOR versus WHILE versus UNTIL versus GOTO loop; 
you might come up with some recipes that covered simple situations, but 
in the long run, the best way to learn to program is also the best way 
to learn proofs: by studying lots of good ones, practicing writing 
simple ones, and reading solutions of ones we assign that you have 
trouble with.
>
> Also, the functions made absolutely no sense.  I have no idea what ::= 
> means and
> the reading came no where close to clarifying the matter.  there is no
> explanation of this new and ill-defined symbol.  What does it mean?
It means "equal by definition."  You can always replace ;:= by = if 
that makes you more comfortable.  This def was omitted by accident in 
the initial version of Notes 1.  It is now defined in footnotes in both 
Notes 1 & 2.  FYI: the symbol := (with one colon) is a standard one for 
definitional equality see 
(http://en.wikipedia.org/wiki/Table_of_mathematical_symbols), but it 
conflcts with common notation for assignment statements in programs, so 
I prefer the double colon version.
> Mark Burroughs

Regards, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307


From adamreed@MIT.EDU Mon Feb 13 18:12:50 2006
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On page 7, section 2.6 explains what it means for a propositional 
formula to be valid.

I'm somewhat confused about the difference between validity and 
soundness. My understanding is that for a proposition to be sound, if 
all the antecedents are true then the conclusion must be true; and 
validity means the conclusion is always true no matter the assignment of 
truth values to the antecedents. But which of these, if any, means that 
you've made a mistake somewhere in the proposition? Should propositions 
be both sound and valid?

From fagin@MIT.EDU Mon Feb 13 19:41:07 2006
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Subject: Notes #2 comments & typos
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Passage: Page 12, definitions

I have no idea how we'll be applying these terms (surjective, etc) in 
class. Before today, I didn't even know what surjective and injective 
meant, let alone that the phrases even existed. Once we start 
applying these ideas, I'd like to have a good deal of practice 
because I know I'm going to have a major problem remembering any of 
these. I had issues with domain and range in high school, and now we 
have "codomain" issues, so I'll probably have difficulty with this 
section. I've seen almost everything else in the notes, but I'm most 
worried about the injection/surjection bit.

Ok, and now my I-have-too-much-free-time typo section for Prof. 
Meyer, #5 being (I think) the most important one:

1. Page 3: "Case 1: Suppose that at least 3 people that did meet x."
         "Suppose that there exist at least..." to make it consistent 
with case 2.
2. Page 4, part 2.1: "There are a couple kinds of assertion one..."
         assertion --> assertions
3. Page 7, top expression: Change the italicized primes to Primes to 
be consistent and clearer.
4. Page 7, part 2.6: split the T with "no matter" - there's no space present.
5. Page 7, the same problem as in Notes 1: this is not DeMorgan's 
Law, it's the Distributive one. I could be wrong, as I'm now 
thoroughly confused about which is which. :-)
6. Page 7, quantier --> quantifier
7. Page 10, top: this could be OK, but you have "Power Set" up top 
and "powerset" right below that
8, maybe: Russell's Paradox, page 15: since we're talking about sets 
inside sets, shouldn't we be using the other symbol (the "C" instead 
of the "E")? I could be wrong, since I don't fully understand the 
Paradox. I just wanted to throw that out there.

-Eddie Fagin 
--=====================_11773028==.ALT
Content-Type: text/html; charset="us-ascii"

<html>
<body>
Passage: Page 12, definitions<br><br>
I have no idea how we'll be applying these terms (surjective, etc) in
class. Before today, I didn't even know what surjective and injective
meant, let alone that the phrases even existed. Once we start applying
these ideas, I'd like to have a good deal of practice because I know I'm
going to have a major problem remembering any of these. I had issues with
domain and range in high school, and now we have &quot;codomain&quot;
issues, so I'll probably have difficulty with this section. I've seen
almost everything else in the notes, but I'm most worried about the
injection/surjection bit.<br><br>
Ok, and now my I-have-too-much-free-time typo section for Prof. Meyer, #5
being (I think) the most important one:<br><br>
1. Page 3: &quot;Case 1: Suppose that at least 3 people that did meet
x.&quot;<br>
<x-tab>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</x-tab>
&quot;Suppose that <i>there exist</i> at least...&quot; to make it
consistent with case 2.<br>
2. Page 4, part 2.1: &quot;There are a couple kinds of assertion
one...&quot;<br>
<x-tab>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</x-tab>assertion
--&gt; assertions<br>
3. Page 7, top expression: Change the italicized <i>primes </i>to Primes
to be consistent and clearer.<br>
4. Page 7, part 2.6: split the T with &quot;no matter&quot; - there's no
space present.<br>
5. Page 7, the same problem as in Notes 1: this is not DeMorgan's Law,
it's the Distributive one. I could be wrong, as I'm now thoroughly
confused about which is which. :-)<br>
6. Page 7, quantier --&gt; quantifier<br>
7. Page 10, top: this could be OK, but you have &quot;Power Set&quot; up
top and &quot;powerset&quot; right below that<br>
8, maybe: Russell's Paradox, page 15: since we're talking about sets
inside sets, shouldn't we be using the other symbol (the &quot;C&quot;
instead of the &quot;E&quot;)? I could be wrong, since I don't fully
understand the Paradox. I just wanted to throw that out there.<br><br>
-Eddie Fagin</body>
</html>

--=====================_11773028==.ALT--


From meyer@imap.theory.csail.mit.edu  Mon Feb 13 22:38:47 2006
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Date: Mon, 13 Feb 2006 22:38:53 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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validity applies to a single formula.  Soundness refers to a deduction 
rule, which consists of a set of formulas (the antecedents that appear 
above the line) along with a single formula (the consequent that appears 
below the line).  The connection between sounds and validity is stated 
in problem 2 at end of Notes 1.

Regards, A.

Adam Reed wrote:

> On page 7, section 2.6 explains what it means for a propositional 
> formula to be valid.
>
> I'm somewhat confused about the difference between validity and 
> soundness. My understanding is that for a proposition to be sound, if 
> all the antecedents are true then the conclusion must be true; and 
> validity means the conclusion is always true no matter the assignment 
> of truth values to the antecedents. But which of these, if any, means 
> that you've made a mistake somewhere in the proposition? Should 
> propositions be both sound and valid?



From meyer@imap.theory.csail.mit.edu  Mon Feb 13 23:06:24 2006
Message-ID: <43F15745.8020204@csail.mit.edu>
Date: Mon, 13 Feb 2006 23:06:29 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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Subject: Re: Notes #2 comments & typos
References: <6.2.3.4.2.20060213191906.01c838c8@hesiod>
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
  <meta content="text/html;charset=ISO-8859-1" http-equiv="Content-Type">
  <title></title>
</head>
<body bgcolor="#ffffff" text="#000000">
Eddie Fagin wrote:
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">Passage: Page 12, definitions<br>
  <br>
I have no idea how we'll be applying these terms (surjective, etc) in
class. Before today, I didn't even know what surjective and injective
meant, let alone that the phrases even existed. Once we start applying
these ideas, I'd like to have a good deal of practice because I know
I'm
going to have a major problem remembering any of these. I had issues
with
domain and range in high school, and now we have "codomain"
issues, so I'll probably have difficulty with this section. I've seen
almost everything else in the notes, but I'm most worried about the
injection/surjection bit.<br>
</blockquote>
injection/surjection will get clear with practice -- and we'll let you
bring a glossary to exams so you don't have to memorize the terms.<br>
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite"><br>
Ok, and now my I-have-too-much-free-time typo section for Prof. Meyer,
#5
being (I think) the most important one:<br>
  <br>
1. Page 3: "Case 1: Suppose that at least 3 people that did meet
x."<br>
  <x-tab>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</x-tab>
"Suppose that <i>there exist</i> at least..." to make it
consistent with case 2.<br>
</blockquote>
revised (slightly differently) in&nbsp; response to your comment<br>
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">2. Page 4, part 2.1: "There are a couple kinds of
assertion
one..."<br>
  <x-tab>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</x-tab>assertion
--&gt; assertions<br>
</blockquote>
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">3. Page 7, top expression: Change the italicized <i>primes
  </i>to Primes
to be consistent and clearer.<br>
</blockquote>
Already fixed (always worth downloading the latest version just before
you read a handout).<br>
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">4. Page 7, part 2.6: split the T with "no matter" -
there's no
space present.<br>
</blockquote>
Already fixed
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">5. Page 7, the same problem as in Notes 1: this is not
DeMorgan's Law,
it's the Distributive one. I could be wrong, as I'm now thoroughly
confused about which is which. :-)<br>
</blockquote>
Already fixed
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">6. Page 7, quantier --&gt; quantifier<br>
</blockquote>
fixed, thanks<br>
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">7. Page 10, top: this could be OK, but you have "Power
Set" up
top and "powerset" right below that<br>
</blockquote>
fixed, thanks
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite">8, maybe: Russell's Paradox, page 15: since we're talking
about sets
inside sets, shouldn't we be using the other symbol (the "C"
instead of the "E")? I could be wrong, since I don't fully
understand the Paradox. I just wanted to throw that out there.<br>
</blockquote>
the little "epsilon" is the right symbol.&nbsp; we'll go over Russell again
on Friday.<br>
<blockquote cite="mid6.2.3.4.2.20060213191906.01c838c8@hesiod"
 type="cite"><br>
-Eddie Fagin</blockquote>
I'm really pleased to have your proofreading comments:.<br>
Thanks A.<br>
</body>
</html>


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Not a problem, I'm happy to help. Before I decided on EECS, I wanted 
to be editor-in-chief of the Times. I guess I never quite gave up that dream.

I printed the reading last week, so I apologize for the redundancy. 
I'll try to keep it up-to-date in the future.

Eddie

At 11:06 PM 2/13/2006, you wrote:
>Eddie Fagin wrote:
>>Passage: Page 12, definitions
>>
>>I have no idea how we'll be applying these terms (surjective, etc) 
>>in class. Before today, I didn't even know what surjective and 
>>injective meant, let alone that the phrases even existed. Once we 
>>start applying these ideas, I'd like to have a good deal of 
>>practice because I know I'm going to have a major problem 
>>remembering any of these. I had issues with domain and range in 
>>high school, and now we have "codomain" issues, so I'll probably 
>>have difficulty with this section. I've seen almost everything else 
>>in the notes, but I'm most worried about the injection/surjection bit.
>injection/surjection will get clear with practice -- and we'll let 
>you bring a glossary to exams so you don't have to memorize the terms.
>>
>>Ok, and now my I-have-too-much-free-time typo section for Prof. 
>>Meyer, #5 being (I think) the most important one:
>>
>>1. Page 3: "Case 1: Suppose that at least 3 people that did meet x."
>>         "Suppose that there exist at least..." to make it 
>> consistent with case 2.
>revised (slightly differently) in  response to your comment
>>2. Page 4, part 2.1: "There are a couple kinds of assertion one..."
>>         assertion --> assertions
>>3. Page 7, top expression: Change the italicized primes to Primes 
>>to be consistent and clearer.
>Already fixed (always worth downloading the latest version just 
>before you read a handout).
>>4. Page 7, part 2.6: split the T with "no matter" - there's no space present.
>Already fixed
>>5. Page 7, the same problem as in Notes 1: this is not DeMorgan's 
>>Law, it's the Distributive one. I could be wrong, as I'm now 
>>thoroughly confused about which is which. :-)
>Already fixed
>>6. Page 7, quantier --> quantifier
>fixed, thanks
>>7. Page 10, top: this could be OK, but you have "Power Set" up top 
>>and "powerset" right below that
>fixed, thanks
>>8, maybe: Russell's Paradox, page 15: since we're talking about 
>>sets inside sets, shouldn't we be using the other symbol (the "C" 
>>instead of the "E")? I could be wrong, since I don't fully 
>>understand the Paradox. I just wanted to throw that out there.
>the little "epsilon" is the right symbol.  we'll go over Russell 
>again on Friday.
>>
>>-Eddie Fagin
>I'm really pleased to have your proofreading comments:.
>Thanks A.

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<html>
<body>
Not a problem, I'm happy to help. Before I decided on EECS, I wanted to
be editor-in-chief of the <u>Times</u>. I guess I never quite gave up
that dream.<br><br>
I printed the reading last week, so I apologize for the redundancy. I'll
try to keep it up-to-date in the future. <br><br>
Eddie<br><br>
At 11:06 PM 2/13/2006, you wrote:<br>
<blockquote type=cite class=cite cite="">Eddie Fagin wrote: <br>
<blockquote type=cite class=cite cite="">Passage: Page 12,
definitions<br><br>
I have no idea how we'll be applying these terms (surjective, etc) in
class. Before today, I didn't even know what surjective and injective
meant, let alone that the phrases even existed. Once we start applying
these ideas, I'd like to have a good deal of practice because I know I'm
going to have a major problem remembering any of these. I had issues with
domain and range in high school, and now we have &quot;codomain&quot;
issues, so I'll probably have difficulty with this section. I've seen
almost everything else in the notes, but I'm most worried about the
injection/surjection bit.</blockquote>injection/surjection will get clear
with practice -- and we'll let you bring a glossary to exams so you don't
have to memorize the terms.<br>
<blockquote type=cite class=cite cite=""><br>
Ok, and now my I-have-too-much-free-time typo section for Prof. Meyer, #5
being (I think) the most important one:<br><br>
1. Page 3: &quot;Case 1: Suppose that at least 3 people that did meet
x.&quot;<br>
<x-tab>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</x-tab>
&quot;Suppose that <i>there exist</i> at least...&quot; to make it
consistent with case 2.</blockquote>revised (slightly differently)
in&nbsp; response to your comment<br>
<blockquote type=cite class=cite cite="">2. Page 4, part 2.1: &quot;There
are a couple kinds of assertion one...&quot;<br>
<x-tab>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</x-tab>assertion
--&gt; assertions<br>
3. Page 7, top expression: Change the italicized <i>primes </i>to Primes
to be consistent and clearer.</blockquote>Already fixed (always worth
downloading the latest version just before you read a handout).<br>
<blockquote type=cite class=cite cite="">4. Page 7, part 2.6: split the T
with &quot;no matter&quot; - there's no space
present.</blockquote>Already fixed <br>
<blockquote type=cite class=cite cite="">5. Page 7, the same problem as
in Notes 1: this is not DeMorgan's Law, it's the Distributive one. I
could be wrong, as I'm now thoroughly confused about which is which.
:-)</blockquote>Already fixed <br>
<blockquote type=cite class=cite cite="">6. Page 7, quantier --&gt;
quantifier</blockquote>fixed, thanks<br>
<blockquote type=cite class=cite cite="">7. Page 10, top: this could be
OK, but you have &quot;Power Set&quot; up top and &quot;powerset&quot;
right below that</blockquote>fixed, thanks <br>
<blockquote type=cite class=cite cite="">8, maybe: Russell's Paradox,
page 15: since we're talking about sets inside sets, shouldn't we be
using the other symbol (the &quot;C&quot; instead of the &quot;E&quot;)?
I could be wrong, since I don't fully understand the Paradox. I just
wanted to throw that out there.</blockquote>the little
&quot;epsilon&quot; is the right symbol.&nbsp; we'll go over Russell
again on Friday.<br>
<blockquote type=cite class=cite cite=""><br>
-Eddie Fagin</blockquote>I'm really pleased to have your proofreading
comments:.<br>
Thanks A.</blockquote></body>
</html>

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From rfh08@MIT.EDU Tue Feb 14 15:23:45 2006
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From: Richard Hermosillo <rfh08@MIT.EDU>
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  One of the more difficult concepts in the reading is exactly how 
changing the order of quantifiers changes the meaning of the 
proposition. I can see how this happens for simpler examples, but it 
would be great if we could go over this a little bit more in the next 
lecture. I think a few more examples and explanations as to how and why 
the meaning of the proposition is changed would really help.

  One more little side note: I like the irony in the third item of the 
website's instructions for these e-mail comments -- "poolry written."

Richard Hermosillo

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3.5: Functions: pp. 11-14

In addition to being annoying ("sur" vs. "bi" vs. "in" as meaningless
prefixes), this section is a little confusing. I'd like to know more about
how we will be using functions. Up until now, my dealings with functions
have been confined to calculus; in addition to the new vocabulary and
symbols (f-hat), what else are they useful for?

~Olga Wichrowska

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First, I believe that section 2.6 (Validity) is missing a "what" in the 
second paragraph, right between "no matter" & "values". I've never before 
heard of the distinction between codomain and range (and I'm not sure I 
understand why the distinction is made) nor seen Russell's paradox in that 
particular notation (since I am already somewhat familiar with it, I 
understand what's going on but it may be worthwhile to write it out more 
explicitly for people who've never seen it before). Also, I don't believe 
it's mentioned before the next-to-last page that these are the 
Zermelo-Fraenkel Axioms, so at least spelling out the acronym in the last 
paragraph would be clearer. Lastly, I was surprised to hear that there are 
collections that are simply "too big" to be sets, especially because I 
don't have any clue what it _means_ to be too big to be a set.
Safia Chettih


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Page 5: The ZFC Axioms
I had a little difficulty in understanding the theorem which is written in terms
of pure mathematical symbols.

From edelmann@MIT.EDU Tue Feb 14 21:58:14 2006
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6.042-staff,

Reading Comments:

At the top of page 3, the reading discusses proving by cases, but it never
provides a clear understanding of how to structure this type of proof.  I know
we briefly went over this in lecture, but I think it should be further
emphasized in lecture about how best to decide what cases to use for the proof.
 The document goes further to an example of using subcases to prove other cases,
and it would be useful if that would covered in class in more depth.

-Nick Edelman




_____________________
Nicholas A. Edelman
Electrical Engineering and Computer Science | 2008
Massachusetts Institute of Technology
3 Ames Street, Box 265
Cambridge, MA 02142
Phone: 617 225 6356

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From: Sean C McDuffee <scmcduff@MIT.EDU>
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Subject: reading comments - Sean McDuffee
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pg 2 ln1 2 Propositions: I thought asking your TA for the complete list of
natural numbers was hilarious

pg 7 ln2 2.4.1 Variables Over One Domain:  I found the goldbach's conjecture
propositional formula confusing to look at.  That particular example detered me
from arranging all the variables to range over one domain.  Probably just me.

pg 7 ln2 2.6 Validity:  ..., a formula now must evaluate to true no matter WHAT
values its...

pg 8 ln2 3 Mathematical Data Types:  Nooo!! Not Tailspin too??!! :(

All in all, I like the notes.  Particularly the side bars like the box on
Russell's Paradox and the SAT problem.  It's nice to see course notes that
branch away from the central material a little bit.  Kinda keeps you thinking
and reading as opposed to just studying.  Only minor grammar problems and
sometimes a word missing really.  Nothing that detracts from the reading.  I
appreciate how quickly you fixed the errors in the first week's notes.

-- 
Sean McDuffee
Research Specialist
MIT-NW17
175 Albany St.
Cambridge, MA 02139 USA
Tel. 617-253-0599
Fax. 617-258-7929
Email: scmcduff@mit.edu

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I really liked the way the section "Quantifying a Predicate" was applied to
a simple situation.   In pgs 5-6,  the application of the phrase "if you can
solve any problem we come up with, then you get an A for the course and goes
through with several examples to display this predicate.   I thought it was
well written.  

 

There is also what seems to be a grammatical error in pg. 8 on the last
paragraph.  The first sentence reads "The same idea extends to predicate
formulas, but to be valid, a formula now must evaluate to true no matter
values."  I believe there should be the word "what" in there.

 




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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>I really liked the way the section &#8220;Quantifying =
a
Predicate&#8221; was applied to a simple situation. &nbsp;&nbsp;In pgs
5-6,&nbsp; the application of the phrase &#8220;if you can solve any =
problem we
come up with, then you get an A for the course and goes through with =
several
examples to display this predicate. &nbsp;&nbsp;I thought it was well
written.&nbsp; <o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>There is also what seems to be a grammatical error in =
pg. 8
on the last paragraph. &nbsp;The first sentence reads &#8220;The same =
idea
extends to predicate formulas, but to be valid, a formula now must =
evaluate to
true no matter values&#8230;&#8221;&nbsp; I believe there should be the =
word &#8220;what&#8221;
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------=_NextPart_000_0006_01C631BB.EBC37B20--


From pat7337@MIT.EDU Tue Feb 14 23:41:01 2006
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Date: Tue, 14 Feb 2006 23:39:28 -0500
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From: Patrick Maher <pat7337@MIT.EDU>
Subject: Week 2 Comments
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I did not really understand the material on functions well.  I am 
also having trouble remembering surjective. injective, etc., but this 
is probably due to unfamiliarity, so I'll get them soon.
    Thanks!
       Pat


From kelleyk@MIT.EDU Wed Feb 15 01:06:49 2006
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Date: Wed, 15 Feb 2006 01:06:39 -0500
From: Kevin Kelley <kelleyk@MIT.EDU>
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I found the discussion of vocabulary for predicates and sets on pp. 
12-13 to be the most difficult part of the reading.  The words 
'surjective' and 'injective' do not suggest a definition to me.  In 
addition, considering the interrelations between total and non-total 
functions and surjective, injective, and bijective functions (i.e., 
which must also have what other properties) confused me for a while, 
although I think that this is also a product of unintuitive vocabulary.  
I think that Prof. Meyer's treatment of the material in class Wednesday 
will probably set this straight in my mind for good.

I also had a little bit of trouble with the discussion of the "for 
every" and "there exists" symbols on pp 7-8 on my first read-through, 
although they gave me less trouble.  The symbols themselves are 
relatively easy to understand, but the various propositions involving 
various combinations of them (i.e., for each X there exists a Y such 
that P(x,y) is true implies that...) required some thought.  It would be 
helpful if these were covered in class.

Kevin

From ruthdhan@MIT.EDU Wed Feb 15 01:43:42 2006
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From: Ruth Dhanaraj <ruthdhan@MIT.EDU>
To: 6042-probs@theory.csail.MIT.EDU
Subject: reading assignment
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Hi,

I think probably the trickiest part of the reading was the order of 
quantifiers section (2.4, page 6). I found that the part of the online 
tutor pset (2.6, where they had four very similar statements, only with 
the quantifiers switched), where the explanation had counterexamples was 
very helpful, and you might want to have that sort of discussion in the 
actual notes.

Ruth Dhanaraj

From lisa_hsu@MIT.EDU Wed Feb 15 01:54:51 2006
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Date: Wed, 15 Feb 2006 01:54:47 -0500
From: Lisa Hsu <lisa_hsu@MIT.EDU>
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I found section 3.5 the most difficult to understand.  I understand it better
now that I have read through it a few more times and completed the online tutor
questions.  However, I think it might have helped to start with the more
colloquial words describing the concept of expressing a function by its domain,
codomain, and set (page 13 "Everything about a function is captured by three
sets..."), rather than with the definition of these concepts (page 11 "A
function assignes an element of one set..."), which made it very abstract.

From veenav@gmail.com Wed Feb 15 02:21:03 2006
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From: Veena <veenav@gmail.com>
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Subject: Reading comments week 2
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Hi,

I haven't worked much with sets before so I found the passage at the
top of page 14 confusing; it would be great if you could explain the
difference between f and f-hat, and the difference between the range
and codomain.  Also, I don't think I really got Russell's Paradox on
page 15.

Thanks!

--Veena Venkatachalam


From irenefan@gmail.com Wed Feb 15 02:30:59 2006
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From: Irene Fan <irenefan@mit.edu>
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On page 11 under section 3.5, I was a little bit confused about the part
that says: "The familiar notation "f (a) = b" indicates that f assigns the
element b $B":(B B to a." From the previous two-sentence explanation of the f:
A->B notation, it seems that function f assigns A to B where A is the domain
and B is the codomain, and although assigning A to B seems to be the same as
assigning b to a, I am not sure whether "a" is the codomain or domain in
f(a) = b.

---------------------------------------------
Irene Fan
irenefan@mit.edu

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On page 11 under section 3.5, I was a little bit confused about the
part that says: &quot;The familiar notation "f (a) = b" indicates that f
assigns the element b $B":(B B to a.&quot; From the previous two-sentence
explanation of the f: A-&gt;B notation, it seems that function f
assigns A to B where A is the domain and B is the codomain, and
although assigning A to B seems to be the same as assigning b to a, I
am not sure whether &quot;a&quot; is the codomain or domain in f(a) = b.<br>
<br>
---------------------------------------------<br>
Irene Fan<br>
<a href="mailto:irenefan@mit.edu">irenefan@mit.edu</a><br>

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From robfalco@MIT.EDU Wed Feb 15 04:02:42 2006
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Date: Wed, 15 Feb 2006 04:02:40 -0500
From: Robert F Falconi <robfalco@MIT.EDU>
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The portion of the reading which I found most difficult was section 3.5 of
Functions starting on page 11. I found the reading itself to be very confusing,
and also felt that it didn't prepare me to do the online tutor question which
involved functions. I am confused the most by the f(A') that appear with a
little carrot over the f and by the ideas of surjective and injective. Could we
please go over this topic more fully in class tomorrow?
-Robert Falconi

From fanyang@MIT.EDU Wed Feb 15 05:01:43 2006
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Name: Fan Yang

Citing a passage:
The passage is at top of page 8 in the "Validity" section.

(I have trouble copying and pasting because there are lots of symbols that MIT
webmail doesn't recognize.)

I think this is the most difficult section for me. I couldn't follow the proof
very well. I wasn't sure what some of the terms, such as "binary predicate", 
meant. I wish the notes could have talked more in detail about the quantifiers
and used an real-life example that varify the truth of the proposition before
going into proving it with quantifiers and logic. I guess another reason could
be that I dont' have much prior experience with the material. I would like to
have testing validity of statements discussed more fully in the next lecture.

From libian@MIT.EDU Wed Feb 15 05:39:44 2006
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Date: Wed, 15 Feb 2006 05:39:36 -0500
From: Li Bian <libian@MIT.EDU>
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Subject: 6.042 reading assignment comments
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Hi,

Thank you for reading the e-mail.

I didn't quite understand page 3; the example of 6 people each other is too
convoluted for demonstrating the "proof by cases" point.  I think I understand
the main point, but the example made me think for a while.

The example on the top of page 6 is a very clear demonstration of the
translation from English to math, and is very interesting.

For proving the two assertions on page 8, I think some concrete examples might
make it clearer.

Question on Page 9:  Does the set of Natural numbers include zero?
Also on Page 9 Section 3.2 2nd Paragraph last line: if set A is either a subset
of A or is A, how come that A does not equal to A?

Thank you.

Li Bian

From pawand@MIT.EDU Wed Feb 15 07:18:34 2006
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From: "Pawan Deedwaniya" <pawand@MIT.EDU>
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Subject: Week 2 Comments
Date: Wed, 15 Feb 2006 07:18:25 -0500
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X-Keywords: NonJunk NotJunk                                                                                   

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Hi, my comments on the reading are:

1.	I feel like truth tables might have been a good addition somewhere.
Although most of the truth tables are pretty straightforward.
2.	I wasn't completely sure what the ZFC axioms were besides that they
were named after a guy named Zermelo. 
3.	The whole f(hat)(A') thing was a little confusing overall. 

 

Pawan Deedwaniya


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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
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are:<o:p></o:p></span></font></p>

<ol style=3D'margin-top:0in' start=3D1 type=3D1>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo1'><font size=3D2 =
face=3DArial><span
     style=3D'font-size:10.0pt;font-family:Arial'>I feel like truth =
tables might
     have been a good addition somewhere. Although most of the truth =
tables are
     pretty straightforward.<o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo1'><font size=3D2 =
face=3DArial><span
     style=3D'font-size:10.0pt;font-family:Arial'>I wasn&#8217;t =
completely sure
     what the ZFC axioms were besides that they were named after a guy =
named
     Zermelo. <o:p></o:p></span></font></li>
 <li class=3DMsoNormal style=3D'mso-list:l0 level1 lfo1'><font size=3D2 =
face=3DArial><span
     style=3D'font-size:10.0pt;font-family:Arial'>The whole =
f(hat)(A&#8217;)
     thing was a little confusing overall. =
<o:p></o:p></span></font></li>
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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>Pawan Deedwaniya<o:p></o:p></span></font></p>

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From abelay@MIT.EDU Wed Feb 15 07:36:39 2006
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Subject: Reading Comments
From: Adam Belay <abelay@MIT.EDU>
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I found the explanation of functions most difficult, especially on the
bottom of page 13 where the function "f-hat" seems to constrain the
domain of the function "f".  I'm curious as to how this concept might be
used.


From costan@MIT.EDU Wed Feb 15 09:17:40 2006
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From: "Victor Costan" <costan@MIT.EDU>
To: <6042-probs@theory.lcs.mit.edu>
Subject: Reading assignment for 02/15
Date: Wed, 15 Feb 2006 09:17:40 -0500
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Here's a suggestion that struck me while reading the lecture notes:

On section 2.5 (page 7), after having the nice explanation about how
quantifiers change if you move the "not" across them", you might want to add
an observation that two "not" quantifiers cancel each other out.

Hope this is good enough for the assignment. I can't think of anything else
that wasn't straightforward in the notes.

	Victor Costan

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From jwan@MIT.EDU Wed Feb 15 09:42:36 2006
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Date: Wed, 15 Feb 2006 09:42:33 -0500
From: Jordan X Wan <jwan@MIT.EDU>
To: 6042-probs@theory.lcs.MIT.EDU
Subject: comments on Week1 Notes
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in regards to page 5 of Week 1 notes:

The ZFC Axioms

I thought the statement "all of mathematics can be derived from these axioms
together with a few logical deduction rules" is quite a bold statement.  I'd
say either show some ways of applying ZFC axioms to non-trivial proofs and show
that it can indeed build many mathematical concepts or take out that passage. 
It seems a little disconnected since in 3.1 it actually states that ZFC is
quite primitive and we will assume a larger set of axioms (from high school
math).  I would like to see some applications of the ZFC axioms.


Jordan

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From: "U.b." <ubong@MIT.EDU>
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Subject: Week 2 Comments
Date: Wed, 15 Feb 2006 10:07:39 -0500
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X-Keywords: NonJunk NotJunk                                                                                   

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On Validity and Soundness:
These two concepts initially seemed somewhat equivalent to me.  I guess
I was a bit confused at first.  However, upon revisiting past lectures,
it turns out they are related but different; a proposition could be
sound but not necessarily valid.  A valid proposition is always sound.
I guess I'm assuming soundness applies not just to propositional
logic/rules but to propositional formulas, and that there is a
difference between a rule and a formula.  In any case, I'll probably
comb through the textbook to see more examples.
 
 
Ubong (U.b.) Ukoh
 

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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
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Soundness:<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
font-family:Arial'>These two concepts initially seemed somewhat =
equivalent to
me.<span style=3D'mso-spacerun:yes'>&nbsp; </span>I guess I was a bit =
confused at
first.<span style=3D'mso-spacerun:yes'>&nbsp; </span>However, upon =
revisiting
past lectures, it turns out they are related but different; a =
proposition could
be sound but not necessarily valid.<span =
style=3D'mso-spacerun:yes'>&nbsp;
</span>A valid proposition is always sound.<span
style=3D'mso-spacerun:yes'>&nbsp; </span>I guess I&#8217;m assuming =
soundness
applies not just <span class=3DGramE>to</span> propositional logic/rules =
but to
propositional formulas, and that there is a difference between a rule =
and a
formula.<span style=3D'mso-spacerun:yes'>&nbsp; </span>In any case, =
I&#8217;ll
probably comb through the textbook to see more =
examples.<o:p></o:p></span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
style=3D'font-size:10.0pt;
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<p class=3DMsoNormal><font size=3D2 face=3DArial><span =
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Ukoh<o:p></o:p></span></font></p>

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From presbrey@MIT.EDU Wed Feb 15 10:15:12 2006
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Date: Wed, 15 Feb 2006 10:15:08 -0500 (EST)
From: Joe <presbrey@MIT.EDU>
To: 6042-probs@theory.csail.MIT.EDU
Subject: lecture notes 2
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The assignment was pretty quick reading.  In fact, it seemed more like a 
sheet of terms with definitions than a reading assignment except for the 
"state of mathematics" comments in part 4 which I found interesting.  One 
thing I was curious about was what Power Sets are used for.  I don't have 
a very good working knowledge of sets and understood the definition but 
didn't see how it could be used.

From meyer@csail.mit.edu Wed Feb 15 10:29:07 2006
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Date: Wed, 15 Feb 2006 10:29:15 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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Subject: Re: Reading Comments for 2/15
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every typo fixed is a help.<br>
thanks, A.<br>
<pre class="moz-signature" cols="72">Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science &amp; AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
<a class="moz-txt-link-abbreviated" href="mailto:meyer@csail.mit.edu">meyer@csail.mit.edu</a>
<a class="moz-txt-link-freetext" href="http://theory.csail.mit.edu/~meyer/">http://theory.csail.mit.edu/~meyer/</a> </pre>
<br>
<br>
Giovanni Reveles wrote:
<blockquote cite="mid200602150410.k1F4AI4h019844@outgoing.mit.edu"
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  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;">I really liked the way
the section &#8220;Quantifying a
Predicate&#8221; was applied to a simple situation. &nbsp;&nbsp;In pgs
5-6,&nbsp; the application of the phrase &#8220;if you can solve any problem we
come up with, then you get an A for the course and goes through with
several
examples to display this predicate. &nbsp;&nbsp;I thought it was well
written.&nbsp; <o:p></o:p></span></font></p>
  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;"><o:p>&nbsp;</o:p></span></font></p>
  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;">There is also what seems
to be a grammatical error in pg. 8
on the last paragraph. &nbsp;The first sentence reads &#8220;The same idea
extends to predicate formulas, but to be valid, a formula now must
evaluate to
true no matter values&#8230;&#8221;&nbsp; I believe there should be the word &#8220;what&#8221;
in there.<o:p></o:p></span></font></p>
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--------------090705020409070101040103--

From meyer@csail.mit.edu Wed Feb 15 10:30:40 2006
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Date: Wed, 15 Feb 2006 10:30:48 -0500
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surj/inj/bij on Friday, but AE/EA today.
regards, A.

Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
meyer@csail.mit.edu
http://theory.csail.mit.edu/~meyer/ 



Kevin Kelley wrote:

> I found the discussion of vocabulary for predicates and sets on pp. 
> 12-13 to be the most difficult part of the reading.  The words 
> 'surjective' and 'injective' do not suggest a definition to me.  In 
> addition, considering the interrelations between total and non-total 
> functions and surjective, injective, and bijective functions (i.e., 
> which must also have what other properties) confused me for a while, 
> although I think that this is also a product of unintuitive 
> vocabulary.  I think that Prof. Meyer's treatment of the material in 
> class Wednesday will probably set this straight in my mind for good.
>
> I also had a little bit of trouble with the discussion of the "for 
> every" and "there exists" symbols on pp 7-8 on my first read-through, 
> although they gave me less trouble.  The symbols themselves are 
> relatively easy to understand, but the various propositions involving 
> various combinations of them (i.e., for each X there exists a Y such 
> that P(x,y) is true implies that...) required some thought.  It would 
> be helpful if these were covered in class.
>
> Kevin


From soroush@MIT.EDU Wed Feb 15 10:35:57 2006
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From: Soroush Vosoughi <soroush@MIT.EDU>
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Subject: Reading assignment
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Hello,
Passage 3.2.2 on page 10 has a defination of the power set. It says that the
power set is the collection of all subsets of a set A. It also says that if A
has n elements then P(A) would have 2^n elements. Why is that? I tried proving
it but couldn't. Could you please prove that in class.
Thank you
-Soroush Vosoughi

From dbw@MIT.EDU Wed Feb 15 10:47:03 2006
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From: Daniel B Wilson <dbw@MIT.EDU>
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Professor Meyer,

The reading was overall rather informative. However, I had trouble when applying
the information about surjective, injective, and bijective to the tutor problem.
I felt like the notes to not completely explain the how |f(A)| relates to |B|
when f is surjective, or injective.

-Daniel

From jessicao@MIT.EDU Wed Feb 15 10:59:58 2006
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I would like to see other examples of why the first assertion in the last part
of section 2.6 ( top of page 8) is valid while the second assertion is not. I
understand in the first assertion that the order of the quantifiers can be
reversed without changing the truthfulness of the predicate.

From wysun07@gmail.com Wed Feb 15 11:46:02 2006
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Date: Wed, 15 Feb 2006 11:45:58 -0500
From: Weiyang Sun <wysun07@gmail.com>
Reply-To: wysun@mit.edu
To: 6042-probs@theory.lcs.mit.edu
Subject: 6.042 Reading
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X-Keywords: NonJunk NotJunk                                                                                   

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I thought that the reading would have made a little more sense, and been a
bit more understandable, if section three had been put at the beginning
(example, the review of mathematical data types on page 8 and the
definitions of surjective, bijective, etc.), because I tried the tutorial
problems after reading each section I thought was associated with it.  Also=
,
I was wondering what the ZFC in ZFC axioms stand for. (pg 14, section 4.)

  - Weiyang Sun

--
"The winner ain't the one with the fastest car, son. It's just the one who
refuses to lose."  --ralph earnhardt

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I thought that the reading would have made a little more sense, and
been a bit more understandable, if section three had been put at the
beginning (example, the review of mathematical data types on page 8 and
the definitions of surjective, bijective, etc.), because I tried the
tutorial problems after reading each section I thought was associated
with it.&nbsp; Also, I was wondering what the ZFC in ZFC axioms stand
for. (pg 14, section 4.)<br>
<br>
&nbsp; - Weiyang Sun<br clear=3D"all"><br>-- <br>&quot;The winner ain't the=
 one with the fastest car, son. It's just the one who refuses to lose.&quot=
;&nbsp;&nbsp;--ralph earnhardt

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From meyer@imap.theory.csail.mit.edu  Wed Feb 15 10:29:07 2006
Message-ID: <43F348CB.1090407@csail.mit.edu>
Date: Wed, 15 Feb 2006 10:29:15 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
Organization: MIT CSAIL
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To: greveles@MIT.EDU
CC: 6042-probs@theory.lcs.mit.edu
Subject: Re: Reading Comments for 2/15
References: <200602150410.k1F4AI4h019844@outgoing.mit.edu>
In-Reply-To: <200602150410.k1F4AI4h019844@outgoing.mit.edu>
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 boundary="------------090705020409070101040103"
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This is a multi-part message in MIME format.
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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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every typo fixed is a help.<br>
thanks, A.<br>
<pre class="moz-signature" cols="72">Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science &amp; AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
<a class="moz-txt-link-abbreviated" href="mailto:meyer@csail.mit.edu">meyer@csail.mit.edu</a>
<a class="moz-txt-link-freetext" href="http://theory.csail.mit.edu/~meyer/">http://theory.csail.mit.edu/~meyer/</a> </pre>
<br>
<br>
Giovanni Reveles wrote:
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  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;">I really liked the way
the section &#8220;Quantifying a
Predicate&#8221; was applied to a simple situation. &nbsp;&nbsp;In pgs
5-6,&nbsp; the application of the phrase &#8220;if you can solve any problem we
come up with, then you get an A for the course and goes through with
several
examples to display this predicate. &nbsp;&nbsp;I thought it was well
written.&nbsp; <o:p></o:p></span></font></p>
  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;"><o:p>&nbsp;</o:p></span></font></p>
  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;">There is also what seems
to be a grammatical error in pg. 8
on the last paragraph. &nbsp;The first sentence reads &#8220;The same idea
extends to predicate formulas, but to be valid, a formula now must
evaluate to
true no matter values&#8230;&#8221;&nbsp; I believe there should be the word &#8220;what&#8221;
in there.<o:p></o:p></span></font></p>
  <p class="MsoNormal"><font face="Arial" size="2"><span
 style="font-size: 10pt; font-family: Arial;"><o:p>&nbsp;</o:p></span></font></p>
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From meyer@imap.theory.csail.mit.edu  Wed Feb 15 10:30:40 2006
Message-ID: <43F34928.1020505@csail.mit.edu>
Date: Wed, 15 Feb 2006 10:30:48 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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To: Kevin Kelley <kelleyk@MIT.EDU>
CC: 6042-probs@theory.csail.mit.edu
Subject: Re: Comments on Reading Assignment for Feb. 15
References: <43F2C4EF.1050201@mit.edu>
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surj/inj/bij on Friday, but AE/EA today.
regards, A.

Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
meyer@csail.mit.edu
http://theory.csail.mit.edu/~meyer/ 



Kevin Kelley wrote:

> I found the discussion of vocabulary for predicates and sets on pp. 
> 12-13 to be the most difficult part of the reading.  The words 
> 'surjective' and 'injective' do not suggest a definition to me.  In 
> addition, considering the interrelations between total and non-total 
> functions and surjective, injective, and bijective functions (i.e., 
> which must also have what other properties) confused me for a while, 
> although I think that this is also a product of unintuitive 
> vocabulary.  I think that Prof. Meyer's treatment of the material in 
> class Wednesday will probably set this straight in my mind for good.
>
> I also had a little bit of trouble with the discussion of the "for 
> every" and "there exists" symbols on pp 7-8 on my first read-through, 
> although they gave me less trouble.  The symbols themselves are 
> relatively easy to understand, but the various propositions involving 
> various combinations of them (i.e., for each X there exists a Y such 
> that P(x,y) is true implies that...) required some thought.  It would 
> be helpful if these were covered in class.
>
> Kevin



From garym@MIT.EDU Wed Feb 15 14:49:48 2006
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From: "Gary M. Matthias" <garym@MIT.EDU>
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On Page 15 of the Week 2 Reading was Russell's Paradox. Maybe I was not
wholly paying attention at the time of reading, but I found it difficult
to understand how exactly it presented a problem. It took three
readings, at three different sittings until a lightbulb finally
activated in my mind. Maybe this can be written more clearly, if this is
at all possible.


Gary Matthias

From estick@MIT.EDU Wed Feb 15 15:10:58 2006
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I would most have liked to learn more about the section on Validity on page 7
(the last paragraph).  I found it very interesting that we could have
conjectures over any possible set and I'd like to hear some examples, because
it seems that they would be rare, given how odd the sets we can choose are. 
We've got {#t, #f}, the natural numbers, or just a set of strings.  It would be
interesting to learn what kinds of logic apply equally well to all of these
things.

-Eli Stickgold

From pengk@MIT.EDU Wed Feb 15 17:03:17 2006
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Apologies for not submitting comments earlier this morning.

Page 5:
I felt the most interesting section in the reading was the snippet about 
the ZFC axioms. Given that I think the fundamentals are an important 
part of a class that is proof-centric as this, I would have liked to see 
them be mentioned in class, despite the fact that they are incredibly 
basic, and perhaps too basic for use in our class.

- Kenny Peng

From varenc@MIT.EDU Wed Feb 15 18:14:45 2006
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hello!

During the reading I felt a bit unclear about the use of universal and 
existential.  The text trys to explain it, but it does a less then 
perfect job of making those clear. However overall it was surprisingly 
readable (or at least more then I expected!)

-Chris

From sbhat@MIT.EDU Wed Feb 15 18:33:59 2006
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From: Sharat Bhat <sbhat@MIT.EDU>
To: 6042-probs@theory.lcs.MIT.EDU
Subject: Predicates and Sets comments
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Sorry about forgetting to send this in the morning.

No one knows whether the ZFC axioms are logically consistent; there is some
possibility that
one person might prove a proposition P and another might prove the proposition
¬P. Then
Math would be broken. This sounds like a crazy situation, but it has happened
before. At the
beginning of the 20th century, the logician Gotlob Frege made an initial attempt
to axiomatize
set theory using a few very plausible axioms. Several mathematicians? most
famously
Bertrand Russell5? discovered that Frege?s axioms actually were
self-contradictory!

I found the section on Russell's Paradox somewhat interesting.  I think I may
have heard of this example earlier, but I had forgotten what exactly the
paradox was.  I would've liked to see more explanation about the difference
between soundness and validity.  Otherwise all the other stuff about proofs,
logic, and sets I already knew about.

-Sharat Bhat


From meyer@csail.mit.edu Wed Feb 15 20:39:19 2006
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From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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I'll be talking about function some on Fri, which I hope will help.

Regards, A.

P.S. In future please send your email reading comments to
    6042-probs@theory.csail.mit.edu
which goes to me AND all the TA's.  You're likelier to get a prompt 
response that way.

Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
meyer@csail.mit.edu
http://theory.csail.mit.edu/~meyer/ 



Helen T Chou wrote:

> Hi Professor Meyer,
>  
> The reading is pretty clear and concise, though I'll admit that the 
> "upside-down A" and "upside-down E" is a bit confusing to handle 
> sometimes, especially when they're given in a certain order. Some of 
> it is a bit non-intuitive to think through, as in the problem sets 
> with the variable sets x and y. Also, functions are confusing - I 
> never really got the hang of functions and functions-within-functions. 
> The reading is clear, though.
>  
> -Helen T Chou


From meyer@imap.theory.csail.mit.edu  Wed Feb 15 12:08:27 2006
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X-Keywords: NonJunk NotJunk                                                                                            

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Hi Professor Meyer,

The reading is pretty clear and concise, though I'll admit that the
"upside-down A" and "upside-down E" is a bit confusing to handle sometimes,
especially when they're given in a certain order. Some of it is a bit
non-intuitive to think through, as in the problem sets with the variable
sets x and y. Also, functions are confusing - I never really got the hang o=
f
functions and functions-within-functions. The reading is clear, though.

-Helen T Chou

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<div>Hi Professor Meyer,</div>
<div>&nbsp;</div>
<div>The reading is pretty clear and concise, though I'll admit that the &q=
uot;upside-down A&quot; and &quot;upside-down E&quot; is a bit confusing to=
 handle sometimes, especially when they're given in a certain order. Some o=
f it is a bit non-intuitive to think through, as in the problem sets with t=
he variable sets x and y. Also, functions are confusing - I never really go=
t the hang of functions and functions-within-functions. The reading is clea=
r, though.
</div>
<div>&nbsp;</div>
<div>-Helen T Chou</div>

------=_Part_16127_7777837.1140023291472--

From meyer@csail.mit.edu Wed Feb 15 21:04:46 2006
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Gary M. Matthias wrote:

>On Page 15 of the Week 2 Reading was Russell's Paradox. Maybe I was not
>wholly paying attention at the time of reading, but I found it difficult
>to understand how exactly it presented a problem. It took three
>readings, at three different sittings until a lightbulb finally
>activated in my mind. Maybe this can be written more clearly, if this is
>at all possible.
>
>
>Gary Matthias
>  
>
This writeup has been revised and polished many times and we thought it 
was clear.  I'm open to making further clarifying revisions, but I'll 
need more identification of what you found difficult before I can work 
any simplifications.

Regards, A.

Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
meyer@csail.mit.edu
http://theory.csail.mit.edu/~meyer/ 


From colinc@MIT.EDU Wed Feb 15 23:13:23 2006
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At the beginning of 2.6, it talks about how propositional formulas with implies
(->) being valid.  I'm a little confused with how an implies statement can be
true.  Is A->B equivalent to B or not(A)?

Colin

From abraxis@MIT.EDU Wed Feb 15 23:48:51 2006
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From: "Volodymyr Boldovskyy" <abraxis@MIT.EDU>
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Subject: email assignment 
Date: Wed, 15 Feb 2006 23:48:49 -0500
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Hello,

I'm sorry for submitting the assignment late, but it's better late then
never.

For the notes 1, and example of proposition 2.2 about n^2+n+41, for me it
was instantly obvious that for n=41 the proposition fails, since every term
of a sum then is divisible by 41. While for 40, its not so obvious to
notice. In the example of DeMorgan's law for sets,(theorem 5.1) the symbols
union and intersection were used, but they were introduced only in notes 2.
In theorem 5.2 there seems to be a typo in the sentence before the last,
which should say "so, this equation holds if and only if every term is
actually 0". In the 7.2 chapter in the sentence before the last there is a
typo in the word "physical".  Overall, notes 1 were very approachable and
well written. 

On notes 2, I got much more confusion, because I've never seen any logic
before. For 3.2.2 it seemed there were too little information presented.
There was a question about powersets in the entrance quiz, but I still
cannot figure out why is this concept important. And the idea of Russell's
paradox is also a little hard to grasp.

Thanks for your attention!

Regards,

Vlad


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<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>Hello,</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>I&#8217;m sorry for submitting the assignment =
late,
but it&#8217;s better late then never.</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>For the notes 1, and example of proposition =
2.2 about
n^2+n+41, for me it was instantly obvious that for n=3D41 the =
proposition fails,
since every term of a sum then is divisible by 41. While for 40, its not =
so
obvious to notice. In the example of DeMorgan&#8217;s law for =
sets,(theorem
5.1) the symbols union and intersection were used, but they were =
introduced
only in notes 2. =9AIn theorem 5.2 there seems to be a typo in the =
sentence before
the last, which should say &#8220;so, this equation holds if and only if =
every
term is actually 0&#8221;. In the 7.2 chapter in the sentence before the =
last
there is a typo in the word &#8220;physical&#8221;.=9A Overall, notes 1 =
were very
approachable and well written. </span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>On notes 2, I got much more confusion, because =
I&#8217;ve
never seen any logic before. For 3.2.2 it seemed there were too little
information presented. There was a question about powersets in the =
entrance
quiz, but I still cannot figure out why is this concept important. And =
the idea
of Russell&#8217;s paradox is also a little hard to =
grasp.</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>Thanks for your attention!</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>Regards,</span></font></p>

<p class=3DMsoNormal><font size=3D2 face=3DArial><span lang=3DEN-US =
style=3D'font-size:
10.0pt;font-family:Arial'>Vlad</span></font></p>

</div>

</body>

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Subject: Reading Comments (I made a really dumb mistake)
From: Robert A Rudd <rudd@MIT.EDU>
To: 6042-probs@theory.lcs.MIT.EDU
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Here are my reading comments for the reading assignment. I also included
below  the reason why they're late. I'm really sorry about this.


Comments:
The passage i'm choosing to cite is the short blurb about Russel's
Paradox at the end of the reading(pg. 15 of 15). I found this to be the
most difficult part of the reading. My main source of concern is I'm not
sure why the problem with the proof is that W is not a set. The thing
that seems most obvious to me is that you couldn't define "S not in S"(I
think that's the correct term).I'm really not clear on why the statement
"..we let S be W is invalid, because S ranges over sets, and W is not a
set" is true.




Reason why I was late:

I accidentally sent it to the wrong address. I logged onto athena after
class Thursday only to see:

The original message was received at Wed, 15 Feb 2006 07:57:53 -0500
(EST)
from OUTGOING-AUTH.MIT.EDU [18.7.22.103]

   ----- The following addresses had permanent fatal errors -----
<042-probs@theory.lcs.MIT.EDU>
I apparently forgot the 6 in 6042. Really sorry about that. I can
forward the e-mail for authenticity if you'd like, however I understand
if I have to take a penalty for this.

Thanks,
Robert Rudd

From meyer@imap.theory.csail.mit.edu  Mon Feb 13 00:45:44 2006
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Fwd: Comments on week 2 notes
Date: Mon, 13 Feb 2006 00:45:44 -0500
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Begin forwarded message:

> From: Albert R. Meyer <meyer@csail.mit.edu>
> Date: February 13, 2006 12:08:45 AM EST
> To: Jorge L De la Garza <mongoose@MIT.EDU>
> Subject: Re: Comments on week 2 notes
>
>
> On Feb 12, 2006, at 3:50 PM, Jorge L De la Garza wrote:
>
>> -p. 8 - proof hard to follow
> yeah, it is pretty abstract and full of symbols
>> -p. 11 - don't understand last line
> It means that to compute f4(T,F), for example, you find the value of 
> [T-->F], which i s F by the rules for implication, and if yuo look at 
> the table, f(T,F) is indeed F.  Does that help?
>
>> -not clear on distinction between a function f and function f with a 
>> hat
> If it remains unclear after rereading. I encourage you to take it up 
> with your TA or me in office hours.
>
> regards, A.
>
> Prof. Albert R. Meyer
> MIT Computer Science & AI Laboratory
> The Stata Center, 32-G624
> 32 Vassar Street
> Cambridge, MA 02139-4307
>

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Begin forwarded message:


<excerpt><bold><color><param>0000,0000,0000</param>From:
</color></bold>Albert R. Meyer <<meyer@csail.mit.edu>

<bold><color><param>0000,0000,0000</param>Date:
</color></bold>February 13, 2006 12:08:45 AM EST

<bold><color><param>0000,0000,0000</param>To: </color></bold>Jorge L
De la Garza <<mongoose@MIT.EDU>

<bold><color><param>0000,0000,0000</param>Subject: </color>Re:
Comments on week 2 notes

</bold>


On Feb 12, 2006, at 3:50 PM, Jorge L De la Garza wrote:


<excerpt>-p. 8 - proof hard to follow

</excerpt>yeah, it is pretty abstract and full of symbols

<excerpt>-p. 11 - don't understand last line

</excerpt>It means that to compute f4(T,F), for example, you find the
value of [T-->F], which i s F by the rules for implication, and if yuo
look at the table, f(T,F) is indeed F.  Does that help?


<excerpt>-not clear on distinction between a function f and function f
with a hat

</excerpt>If it remains unclear after rereading. I encourage you to
take it up with your TA or me in office hours.


regards, A.


Prof. Albert R. Meyer

MIT Computer Science & AI Laboratory

The Stata Center, 32-G624

32 Vassar Street

Cambridge, MA 02139-4307


</excerpt>
--Apple-Mail-1-441071473--


From meyer@imap.theory.csail.mit.edu  Mon Feb 13 00:08:45 2006
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Re: Comments on week 2 notes
Date: Mon, 13 Feb 2006 00:08:45 -0500
To: Jorge L De la Garza <mongoose@MIT.EDU>
Status: RO
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X-Keywords: Forwarded                                                                                                    


On Feb 12, 2006, at 3:50 PM, Jorge L De la Garza wrote:

> -p. 8 - proof hard to follow
yeah, it is pretty abstract and full of symbols
> -p. 11 - don't understand last line
It means that to compute f4(T,F), for example, you find the value of 
[T-->F], which i s F by the rules for implication, and if yuo look at 
the table, f(T,F) is indeed F.  Does that help?

> -not clear on distinction between a function f and function f with a 
> hat
If it remains unclear after rereading. I encourage you to take it up 
with your TA or me in office hours.

regards, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307


From meyer@imap.theory.csail.mit.edu  Sat Feb 11 16:23:29 2006
In-Reply-To: <7b0264ba0602081810r6d8058c3ka08e7441639b1172@mail.gmail.com>
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Re: ln2 comments for reading
Date: Sat, 11 Feb 2006 16:23:29 -0500
To: lee-kai <randomly@mit.edu>
Status: RO
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On Feb 8, 2006, at 9:10 PM, lee-kai wrote:

> I think there should be a period at the end of "you can solve at least 
> one problem we come up with" on page 5.
yes; added.
>
>  Maybe also after "$B"O(Bx $B":(B D. P (n)" and "$B"P(Bx $B":(B D. P (n)".  There are 
> probably some other places that could use them (like the bottom of 
> page 6), but finding places for periods is boring.
maybe, but here I prefer not to have the final periods.

>
>  I found the section about the consistency of ZFC axioms and about the 
> Banach-Tarski Theorem the most interesting!

Thanks, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307


From meyer@imap.theory.csail.mit.edu  Sat Feb 11 16:15:26 2006
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Re: Reading Assignment
Date: Sat, 11 Feb 2006 16:15:26 -0500
To: Richard Hermosillo <rfh08@MIT.EDU>
Status: RO
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On Feb 9, 2006, at 11:06 PM, Richard Hermosillo wrote:

> The most surprising part of the reading was that proving 2 + 2 = 4 
> requires more than 20,000 steps. That's a lot more than I would have 
> expected (I would have thought a lot more than just, say, 2 or 3 
> steps, but far less than 20,000). I'm also curious as to how the 
> figure of 20,000 was reached.

I'm not sure I believe the 20,000 figure myself, but I have been told 
reliably that the 2+2=4 proof does not appear until several hundred 
pages into Principia Mathematica.

regards, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307
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On Feb 9, 2006, at 11:06 PM, Richard Hermosillo wrote:


<excerpt>The most surprising part of the reading was that proving 2 +
2 = 4 requires more than 20,000 steps. That's a lot more than I would
have expected (I would have thought a lot more than just, say, 2 or 3
steps, but far less than 20,000). I'm also curious as to how the
figure of 20,000 was reached.

</excerpt>

I'm not sure I believe the 20,000 figure myself, but I have been told
reliably that the 2+2=4 proof does not appear until several hundred
pages into <italic>Principia Mathematica.

</italic>

regards, A.


Prof. Albert R. Meyer

MIT Computer Science & AI Laboratory

The Stata Center, 32-G624

32 Vassar Street

Cambridge, MA 02139-4307
--Apple-Mail-1-324053101--


From meyer@imap.theory.csail.mit.edu  Sat Feb 11 16:12:42 2006
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Re: Week 1 reading
Date: Sat, 11 Feb 2006 16:12:42 -0500
To: Daniel B Wilson <dbw@MIT.EDU>
Status: RO
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On Feb 10, 2006, at 10:36 AM, Daniel B Wilson wrote:

> Professor Meyer,
>
> I found the reading interesting. I do have a question about an example 
> you used
> regarding Iff statements. The second example used states:
>
> "Two triangles have the same side lengths if and only if all angles 
> are the
> same."
>
> While I agree this is a true statement, I am not sure that the reverse 
> is always
> true. Two triangles with the same angles are similar, but not 
> necessarily
> congruent.
YES, same angles impiles similar, not congruent.  BUT that means the 
"iff" is FALSE: it's only an IMPLIES.
I've fixed it in the notes so it is an IFF along the lines you 
suggested.
>
> My first question is, should the statement read something like "Two 
> triangles
> have the same side lengths if and only if all angles are the same and 
> one side
> is equal." or something thereabouts? Or is it fine the way it is.
>
> My second question is if a statement is not necessarily correct in the 
> reverse,
> is it an Iff?
IFF means true in both directions.

> Otherwise, I found the reading rather informative and appreciated the 
> use or
> common language when able (as is necessary for a good proof).
>
> -Daniel


From meyer@imap.theory.csail.mit.edu  Sat Feb 11 15:47:02 2006
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From: Albert R. Meyer <meyer@csail.mit.edu>
Subject: Notes 1 typos
Date: Sat, 11 Feb 2006 15:47:02 -0500
To: fagin@mit.edu
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Thanks for pointing out the several you found.  Good work -- and keep 
it up!

Regards, A.

Prof. Albert R. Meyer
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139-4307


From gjw@theory.csail.mit.edu Thu Feb 16 16:22:20 2006
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To: Daniel B Wilson <dbw@MIT.EDU>
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X-Keywords: NotJunk                                                                                           

Hi,

On Fri, 10 Feb 2006, Daniel B Wilson wrote:

> I found the reading interesting. I do have a question about an example
> you used regarding Iff statements. The second example used states:
>
> "Two triangles have the same side lengths if and only if all angles are
> the same."
>
> While I agree this is a true statement, I am not sure that the reverse
> is always true. Two triangles with the same angles are similar, but not
> necessarily congruent.

Actually, this is not a true statement -- when we say if and only if, we
mean *both* directions.

> My first question is, should the statement read something like "Two
> triangles have the same side lengths if and only if all angles are the
> same and one side is equal." or something thereabouts? Or is it fine the
> way it is.

It's not fine the way it is -- I believe Albert has fixed it.

> My second question is if a statement is not necessarily correct in the
> reverse, is it an Iff?

The statement is not *true* as an if and only if.  I can write down any
statement that is an if and only if:

I am God if and only if it is sunny today.

This is an "IFF" statement.  Its truth has nothing to do with the fact
that it is an "if and only if" statement.  We can ask if the statement is
true or not, like any statement.

> Otherwise, I found the reading rather informative and appreciated the
> use or common language when able (as is necessary for a good proof).

Sounds good!

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:27:18 2006
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X-Keywords: NotJunk                                                                                           

Hi,

Actually, I don't know too much about this -- but I do know it follows
from axioms from set theory.  The main reason it takes so many steps is
that numbers themselves are not part of set theory -- one constructs
natural numbers from axioms on set theory and then derives theorems about
numbers from the basic axioms of set theory.  This is why a proof of it
was on the order of 20,000 lines -- somebody proved it using this many
lines.  This doesn't mean a shorter proof doesn't exist, but my guess is
that 10,000 of those lines goes towards constructing natural numbers, so a
huge number of lines is needed, anyway.

-Grant

On Thu, 9 Feb 2006, Richard Hermosillo wrote:

> The most surprising part of the reading was that proving 2 + 2 = 4
> requires more than 20,000 steps. That's a lot more than I would have
> expected (I would have thought a lot more than just, say, 2 or 3 steps,
> but far less than 20,000). I'm also curious as to how the figure of
> 20,000 was reached.
>

From gjw@theory.csail.mit.edu Thu Feb 16 16:28:32 2006
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X-Keywords: NotJunk                                                                                           

Hi,

On Tue, 14 Feb 2006, Richard Hermosillo wrote:

>   One of the more difficult concepts in the reading is exactly how
> changing the order of quantifiers changes the meaning of the
> proposition. I can see how this happens for simpler examples, but it
> would be great if we could go over this a little bit more in the next
> lecture. I think a few more examples and explanations as to how and why
> the meaning of the proposition is changed would really help.

I think we did go over this a bit on Wednesday.  Let me know if you want
more help with it.

>   One more little side note: I like the irony in the third item of the
> website's instructions for these e-mail comments -- "poolry written."

Good catch!

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:29:18 2006
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X-Keywords: NotJunk                                                                                           

Hi,

On Tue, 14 Feb 2006, Tural Badirkhanli wrote:

> Page 5: The ZFC Axioms I had a little difficulty in understanding the
> theorem which is written in terms of pure mathematical symbols.

How far did you get?  Did you understand the first few?

They are pretty succinct and hard to follow, but a good exercise for
convincing yourself you understand the exists and forall symbols.

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:32:06 2006
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X-Keywords: NotJunk                                                                                           

Hi,

On Tue, 14 Feb 2006, Sean C McDuffee wrote:

> pg 7 ln2 2.4.1 Variables Over One Domain:  I found the goldbach's
> conjecture propositional formula confusing to look at.  That particular
> example detered me from arranging all the variables to range over one
> domain.  Probably just me.

Let me know if you want more help with this -- what was confusing in
particular?

> pg 7 ln2 2.6 Validity:  ..., a formula now must evaluate to true no
> matter WHAT values its...

I'm not sure what you're getting at here -- is there something in here
that confused you, or are you pointing out a typo that I'm missing?

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:34:57 2006
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Hi,

On Wed, 15 Feb 2006, Lisa Hsu wrote:

> I found section 3.5 the most difficult to understand.  I understand it
> better now that I have read through it a few more times and completed
> the online tutor questions.  However, I think it might have helped to
> start with the more colloquial words describing the concept of
> expressing a function by its domain, codomain, and set (page 13
> "Everything about a function is captured by three sets..."), rather than
> with the definition of these concepts (page 11 "A function assignes an
> element of one set..."), which made it very abstract.

I agree, sometimes it helps to be more colloquial in first describing
something, and then giving the more abstract, precise definition.

Regarding office hours -- I don't think I'm going to change them just yet,
because 4-5 seems a better time for other people (fewer classes occuring
between 4-5pm).  However, I don't mind showing up 30 minutes before class
to help you with some stuff, if you're interested.  Let me know.

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:39:09 2006
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X-Keywords: NotJunk                                                                                           

Hi,

On Wed, 15 Feb 2006, Adam Belay wrote:

> I found the explanation of functions most difficult, especially on the
> bottom of page 13 where the function "f-hat" seems to constrain the
> domain of the function "f".  I'm curious as to how this concept might be
> used.

So, f maps elements of A to elements of B.  I'll be concrete -- let's say
f is the grades function in this class.  A is the set of students, and B
is the natural numbers.  So, for example:

  f(Adam Belay) = A

f-hat, on the other hand, maps *subsets* of students to *subsets* of
grades.  For instance:

  f({Adam Belay, Daniel Wilson, Colin Clarke, Joe Presbrey}) = {A, B}

meaning that everybody in your group has either an A or B in the class.
You can see why we'd be interested in subsets -- for instance, we might
want to know what:

  f(Set of sophomores)

is compared to:

  f(Set of seniors)

Hope this helps!

-Grant

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Subject: Re: comments
From: Tural Badirkhanli <turalb@MIT.EDU>
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X-Keywords: NonJunk NotJunk                                                                                   

Yea, I am pretty comfortable with exists and forall symbols. I okay so
far.


On Thu, 2006-02-16 at 16:29 -0500, Grant Wang wrote:
> Hi,
> 
> On Tue, 14 Feb 2006, Tural Badirkhanli wrote:
> 
> > Page 5: The ZFC Axioms I had a little difficulty in understanding the
> > theorem which is written in terms of pure mathematical symbols.
> 
> How far did you get?  Did you understand the first few?
> 
> They are pretty succinct and hard to follow, but a good exercise for
> convincing yourself you understand the exists and forall symbols.
> 
> -Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:42:55 2006
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Hi,

On Wed, 15 Feb 2006, Jordan X Wan wrote:

> The ZFC Axioms
>
> I thought the statement "all of mathematics can be derived from these
> axioms together with a few logical deduction rules" is quite a bold
> statement.

It is a bold statement.  Indeed, this hasn't been done, but it's believed
to be true.  You suggest that, without hard evidence, we should take it
out of the passage.  I actually disagree -- I think that sometimes
statements like that can be quite amazing and interesting to students.
Furthermore, it expresses the power of the ZFC axioms -- and indeed,
showing this power would be a bit cumbersome.

For instance, if you've taken a hardware course, the analog of the ZFC
axioms might be the instruction set of your computer.  The fact that
*every* program is essentially just the instruction set is pretty amazing.

But thanks for the comments.

-Grant


From gjw@theory.csail.mit.edu Thu Feb 16 16:44:13 2006
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Hi,

On Wed, 15 Feb 2006, Joe wrote:

> The assignment was pretty quick reading.  In fact, it seemed more like a
> sheet of terms with definitions than a reading assignment except for the
> "state of mathematics" comments in part 4 which I found interesting.  One
> thing I was curious about was what Power Sets are used for.  I don't have
> a very good working knowledge of sets and understood the definition but
> didn't see how it could be used.

Power sets are useful when we are trying to describe all possibilities --
for instance, if S is the set of 6.042 students, then P(S), the power set
of S, consists of all subsets of 6.042 students, or all the possible
groups of 6.042 students.  So it can be useful to describe all possible
scenarios.

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:44:47 2006
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X-Keywords: NotJunk                                                                                           

Hi,

We'll see more of this on Friday -- if it still troubles you a bit, let me
know.

-Grant

On Wed, 15 Feb 2006, Daniel B Wilson wrote:

> Professor Meyer,
>
> The reading was overall rather informative. However, I had trouble when applying
> the information about surjective, injective, and bijective to the tutor problem.
> I felt like the notes to not completely explain the how |f(A)| relates to |B|
> when f is surjective, or injective.
>
> -Daniel
>

From gjw@theory.csail.mit.edu Thu Feb 16 16:48:45 2006
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X-Keywords: NotJunk                                                                                           

Hi,

On Wed, 15 Feb 2006 jessicao@MIT.EDU wrote:

> I would like to see other examples of why the first assertion in the
> last part of section 2.6 ( top of page 8) is valid while the second
> assertion is not. I understand in the first assertion that the order of
> the quantifiers can be reversed without changing the truthfulness of the
> predicate.

The best way to see this is to assign meaning to the predicate P -- let's
take it to be:

P(x,y) = x and y are related.

So, the first assertion states:

If there exists an x, such that for all y, x and y are related, then
for every y, there exists an x such that x and y are related.

Say Bob is "x", i.e. he is the person that everybody is related to.  Then
it's definitely the case that, everybody is related to someone, in
particular, Bob!

On the other hand, the second statement says:

If for every y, there is an x such that x and y are related, then there
exists an x, such that for every y, x is related to y.

If everybody is related to someone, is it the case that there is a
*single* person that everybody is related to?  No...I'm only related to my
family, you're only related to your family, etc.  So there's no *one*
person who is related to everybody.

Does that help?

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:49:33 2006
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Hi,

We'll look into it -- it is a bit tricky to understand (at least it was
for me the first time I saw it).  Let me know if you have any suggestions
for rewriting it more clearly.

-Grant

On Wed, 15 Feb 2006, Gary M. Matthias wrote:

> On Page 15 of the Week 2 Reading was Russell's Paradox. Maybe I was not
> wholly paying attention at the time of reading, but I found it difficult
> to understand how exactly it presented a problem. It took three
> readings, at three different sittings until a lightbulb finally
> activated in my mind. Maybe this can be written more clearly, if this is
> at all possible.
>
>
> Gary Matthias
>

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X-Keywords: NotJunk                                                                                           

Hi,

On Wed, 15 Feb 2006, Kenny Peng wrote:

> Apologies for not submitting comments earlier this morning.
>
> Page 5: I felt the most interesting section in the reading was the
> snippet about the ZFC axioms. Given that I think the fundamentals are an
> important part of a class that is proof-centric as this, I would have
> liked to see them be mentioned in class, despite the fact that they are
> incredibly basic, and perhaps too basic for use in our class.

Yes, unfortunately, I think it might be a waste of class-time to talk more
about the ZFC axioms.  However, Albert is quite the expert on foundations
of logic, etc. so if you find yourself curious about something, feel free
to ask him!

-Grant

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X-Keywords: NotJunk                                                                                           

Hi,

On Wed, 15 Feb 2006, Chris Varenhorst wrote:

> During the reading I felt a bit unclear about the use of universal and
> existential.  The text trys to explain it, but it does a less then
> perfect job of making those clear. However overall it was surprisingly
> readable (or at least more then I expected!)

Let me know if you need more help with the universal and existential
quantifiers.  I hope class on Wednesday was helpful.

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:54:10 2006
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X-Keywords: NotJunk                                                                                           

Hi,

On Wed, 15 Feb 2006, Colin M Clarke wrote:

> At the beginning of 2.6, it talks about how propositional formulas with
> implies (->) being valid.  I'm a little confused with how an implies
> statement can be true.  Is A->B equivalent to B or not(A)?

Have you taken a look at the truth table of A->B and written it out?  That
will be the ultimate evidence of the equivalence of A->B and (B or
not(A)).  You are indeed correct, though, that they are equivalent.  Do
you know of a quick explanation why this is the case?

-Grant

From gjw@theory.csail.mit.edu Thu Feb 16 16:57:51 2006
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To: Volodymyr Boldovskyy <abraxis@MIT.EDU>
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X-Keywords: NotJunk                                                                                           

Hi,

On Wed, 15 Feb 2006, Volodymyr Boldovskyy wrote:

> Hello,
>
> I'm sorry for submitting the assignment late, but it's better late then
> never.
>
> For the notes 1, and example of proposition 2.2 about n^2+n+41, for me it
> was instantly obvious that for n=41 the proposition fails, since every term
> of a sum then is divisible by 41. While for 40, its not so obvious to
> notice. In the example of DeMorgan's law for sets,(theorem 5.1) the symbols
> union and intersection were used, but they were introduced only in notes 2.
> In theorem 5.2 there seems to be a typo in the sentence before the last,
> which should say "so, this equation holds if and only if every term is
> actually 0". In the 7.2 chapter in the sentence before the last there is a
> typo in the word "physical".  Overall, notes 1 were very approachable and
> well written.

Good that you saw that the proposition fails for n = 41.  It's an
interesting example of a polynomial where applying it to a few numbers
gives prime numbers, which is sort of cute.  Thanks for the typos, too.

> On notes 2, I got much more confusion, because I've never seen any logic
> before. For 3.2.2 it seemed there were too little information presented.
> There was a question about powersets in the entrance quiz, but I still
> cannot figure out why is this concept important. And the idea of
> Russell's paradox is also a little hard to grasp.

Yes, Russell's paradox is a little hard to grasp.  Indeed, that was more
for informational purposes than being something we think is necessary to
learn.  Let me know if you have more questions about logic.  Regarding
powersets -- they're roughly used when we want to consider all subsets of
a set.  So, if S is the set of 6.042 students, then P(S) is the powerset,
and is the set of all *subsets* of 6.042 students, i.e. all possible
groups of 6.042 students.  It might be useful sometimes to talk about
this, i.e. when trying to figure out how many groups there could possibly
be.

-Grant

From xiaotao88@yahoo.com Thu Feb 16 23:24:45 2006
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Date: Thu, 16 Feb 2006 20:24:39 -0800 (PST)
From: Zuoyu Tao <xiaotao88@yahoo.com>
Subject: Reading comments
To: 6042-probs@theory.lcs.mit.edu
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X-Keywords: NonJunk NotJunk                                                                                   

Hi,

The part I found most surprising was about the Banach
Tarski Theorem. I have heard about the general idea of
the theorem, but I was unaware of the exact pieces (6)
the original ball had to be divided in order to
construct 2 new balls. What significance does the
number 6 have? Why is it not 7 or 5?

Zuoyu Tao

__________________________________________________
Do You Yahoo!?
Tired of spam?  Yahoo! Mail has the best spam protection around 
http://mail.yahoo.com 

From meyer@imap.theory.csail.mit.edu  Wed Feb 15 20:39:19 2006
Message-ID: <43F3D7CF.3090708@csail.mit.edu>
Date: Wed, 15 Feb 2006 20:39:27 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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Subject: Re: 6.042
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I'll be talking about function some on Fri, which I hope will help.

Regards, A.

P.S. In future please send your email reading comments to
    6042-probs@theory.csail.mit.edu
which goes to me AND all the TA's.  You're likelier to get a prompt 
response that way.

Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
meyer@csail.mit.edu
http://theory.csail.mit.edu/~meyer/ 



Helen T Chou wrote:

> Hi Professor Meyer,
>  
> The reading is pretty clear and concise, though I'll admit that the 
> "upside-down A" and "upside-down E" is a bit confusing to handle 
> sometimes, especially when they're given in a certain order. Some of 
> it is a bit non-intuitive to think through, as in the problem sets 
> with the variable sets x and y. Also, functions are confusing - I 
> never really got the hang of functions and functions-within-functions. 
> The reading is clear, though.
>  
> -Helen T Chou



From meyer@imap.theory.csail.mit.edu  Wed Feb 15 21:04:47 2006
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Date: Wed, 15 Feb 2006 21:04:54 -0500
From: "Prof. Albert R. Meyer" <meyer@csail.mit.edu>
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Subject: Re: Comments for Reading...
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Gary M. Matthias wrote:

>On Page 15 of the Week 2 Reading was Russell's Paradox. Maybe I was not
>wholly paying attention at the time of reading, but I found it difficult
>to understand how exactly it presented a problem. It took three
>readings, at three different sittings until a lightbulb finally
>activated in my mind. Maybe this can be written more clearly, if this is
>at all possible.
>
>
>Gary Matthias
>  
>
This writeup has been revised and polished many times and we thought it 
was clear.  I'm open to making further clarifying revisions, but I'll 
need more identification of what you found difficult before I can work 
any simplifications.

Regards, A.

Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science & AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
meyer@csail.mit.edu
http://theory.csail.mit.edu/~meyer/ 



From sjcates@theory.csail.mit.edu Fri Feb 17 12:56:16 2006
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Subject: Re: Reading assignment #1
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Hi Mark,
Good comments.  I see your points.  Maybe it would be helpful to think 
of writing a good proof as an art that takes some practice.  Not very 
satisfying, I know, but it will get easier. 

Sonya

Mark D Burroughs wrote:
> First of all I would like to state that not enough was said about "implies" and
> -> which became exceptionally frustrating on the online problem set.  I had
> thought that the final part on the first problem required a -> since "getting
> an A in the class" was dependent on "getting an A on the final" and "doing all
> the questions".
>
> In the reading itself, it was not clear what would we should be doing with
> proofs.  Moreover, when we should be using proof by contradiction and when we
> should use direct proof.  Obviously direct proof is desired more highly than
> PBC, but when should we not try to use direct proof?  What should we be looking
> for when trying to do a proof, ie what should we be looking for to know to do it
> with direct proof or PBC?  I think a little more clarification on that would be
> helpful.
>
> Also, the functions made absolutely no sense.  I have no idea what ::= means and
> the reading came no where close to clarifying the matter.  there is no
> explanation of this new and ill-defined symbol.  What does it mean?
>
> Mark Burroughs
>   



From sjcates@theory.csail.mit.edu Fri Feb 17 12:57:45 2006
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X-Keywords: NonJunk                                                                                           

Hi Adam,
This was a good question to bring up, I think the explanation in class 
clarified the difference for others too.

Sonya

Adam Reed wrote:
> On page 7, section 2.6 explains what it means for a propositional 
> formula to be valid.
>
> I'm somewhat confused about the difference between validity and 
> soundness. My understanding is that for a proposition to be sound, if 
> all the antecedents are true then the conclusion must be true; and 
> validity means the conclusion is always true no matter the assignment 
> of truth values to the antecedents. But which of these, if any, means 
> that you've made a mistake somewhere in the proposition? Should 
> propositions be both sound and valid?



From sjcates@theory.csail.mit.edu Fri Feb 17 13:21:49 2006
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Hi Olga,
That's a good question and one that gets to the root of what math is.  
One way to think of this might be that functions are one tool that we 
use to model the real world (Think of the solar system example from 
lecture.  It's not math, but we can model it.). So learning about 
functions, their properties, and how to manipulate them will help us to 
make better models and to better understand what we're modeling.  That's 
a very broad answer.  Let me know if your question is really more 
specific than that.

Sonya


Olga Wichrowska wrote:
> 3.5: Functions: pp. 11-14
>
> In addition to being annoying ("sur" vs. "bi" vs. "in" as meaningless
> prefixes), this section is a little confusing. I'd like to know more about
> how we will be using functions. Up until now, my dealings with functions
> have been confined to calculus; in addition to the new vocabulary and
> symbols (f-hat), what else are they useful for?
>
> ~Olga Wichrowska
>
>   



From sjcates@theory.csail.mit.edu Fri Feb 17 14:08:21 2006
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Hi Kevin,

I think these words were coined back when latin or french was the 
universal scientific language. Now that not everyone knows latin, the 
meanings of some terms have lost their intuitiveness.  In this case, it 
might help to know that sur means "on to".

If after today's lecture, things aren't clear, let me know.

Sonya


Kevin Kelley wrote:
> I found the discussion of vocabulary for predicates and sets on pp. 
> 12-13 to be the most difficult part of the reading.  The words 
> 'surjective' and 'injective' do not suggest a definition to me.  In 
> addition, considering the interrelations between total and non-total 
> functions and surjective, injective, and bijective functions (i.e., 
> which must also have what other properties) confused me for a while, 
> although I think that this is also a product of unintuitive 
> vocabulary.  I think that Prof. Meyer's treatment of the material in 
> class Wednesday will probably set this straight in my mind for good.
>
> I also had a little bit of trouble with the discussion of the "for 
> every" and "there exists" symbols on pp 7-8 on my first read-through, 
> although they gave me less trouble.  The symbols themselves are 
> relatively easy to understand, but the various propositions involving 
> various combinations of them (i.e., for each X there exists a Y such 
> that P(x,y) is true implies that...) required some thought.  It would 
> be helpful if these were covered in class.
>
> Kevin



From sjcates@theory.csail.mit.edu Fri Feb 17 16:45:57 2006
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X-Keywords: NonJunk                                                                                           

Hi Fan,
A binary predicate is a function that takes two inputs and returns the 
outcome of some test, either true or false.  I don't think that term was 
used in class, but we did see some examples in class without using that 
name.  An example would be P(x, y) where P returns true if x=y and false 
otherwise, for x, y real numbers.

Did the explanation of validity and soundness that was given in class 
answer your questions?

Sonya

fanyang@MIT.EDU wrote:
> Name: Fan Yang
>
> Citing a passage:
> The passage is at top of page 8 in the "Validity" section.
>
> (I have trouble copying and pasting because there are lots of symbols that MIT
> webmail doesn't recognize.)
>
> I think this is the most difficult section for me. I couldn't follow the proof
> very well. I wasn't sure what some of the terms, such as "binary predicate", 
> meant. I wish the notes could have talked more in detail about the quantifiers
> and used an real-life example that varify the truth of the proposition before
> going into proving it with quantifiers and logic. I guess another reason could
> be that I dont' have much prior experience with the material. I would like to
> have testing validity of statements discussed more fully in the next lecture.
>   



From sjcates@theory.csail.mit.edu Fri Feb 17 17:06:04 2006
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Hi Pawan,
Maybe the ZFC axioms are a bit clearer after today's lecture. That is an 
important and interesting bit of math history, but you won't see the 
axiom of choice on any problems or exams for this class.

I think the three points under Lemma (Mapping Rule) on page 13 of week 2 
are good to know for this class though. Is this the section you were 
referring to on point 3? Here's an example of the difference between f 
and f(hat) that's referred to in that section. Let f be a function on 
the real numbers, f(x)=x^3. So when I write f(x) I mean a single value, 
for example, f(2)=8. If I want to express all the values that the 
function f can have when x is between 1 and 2, I could write f(hat)( 
[1,2] ), which would be the same as the interval [1,8].

Sonya

Pawan Deedwaniya wrote:
>
> Hi, my comments on the reading are:
>
>    1. I feel like truth tables might have been a good addition
>       somewhere. Although most of the truth tables are pretty
>       straightforward.
>    2. I wasn’t completely sure what the ZFC axioms were besides that
>       they were named after a guy named Zermelo.
>    3. The whole f(hat)(A’) thing was a little confusing overall.
>
> Pawan Deedwaniya
>



From sjcates@theory.csail.mit.edu Fri Feb 17 17:33:53 2006
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Hi U. b.,

This is an important distinction. I'm going to quote Professor Meyer on 
this one:

"validity applies to a single formula. Soundness refers to a deduction 
rule, which consists of a set of formulas (the antecedents that appear 
above the line) along with a single formula (the consequent that appears 
below the line). The connection between sounds and validity is stated in 
problem 2 at end of Notes 1. "

Sonya


U.b. wrote:
>
> On Validity and Soundness:
>
> These two concepts initially seemed somewhat equivalent to me. I guess 
> I was a bit confused at first. However, upon revisiting past lectures, 
> it turns out they are related but different; a proposition could be 
> sound but not necessarily valid. A valid proposition is always sound. 
> I guess I’m assuming soundness applies not just to propositional 
> logic/rules but to propositional formulas, and that there is a 
> difference between a rule and a formula. In any case, I’ll probably 
> comb through the textbook to see more examples.
>
> Ubong (U.b.) Ukoh
>



From sjcates@theory.csail.mit.edu Fri Feb 17 17:51:42 2006
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Hi Soroush,
One way of thinking about this would be to assign each element of A 
either a 0 or a 1.  For any subset of A, 0 means that element is not a 
member of the subset and 1 means that it is.  So there would be a 
distinct binary string of length n for every possible subset of A.  
There are 2^n binary strings of length n (do you know why this is true?  
if not ask me.) so there are also 2^n possible subsets of A.  Therefore 
the power set, which is defined as all possible subsets, has 2^n elements.

Here's an example:
A={1,2,3}
subset   string
1             100
2             010
3             001
{1,2}      110
{1,3}      101
{2,3}      011
{1,2,3}   111
emptyset 000


Sonya

Soroush Vosoughi wrote:
> Hello,
> Passage 3.2.2 on page 10 has a defination of the power set. It says that the
> power set is the collection of all subsets of a set A. It also says that if A
> has n elements then P(A) would have 2^n elements. Why is that? I tried proving
> it but couldn't. Could you please prove that in class.
> Thank you
> -Soroush Vosoughi
>   



From sjcates@theory.csail.mit.edu Fri Feb 17 17:57:24 2006
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Was your question answered in class today?

Sonya

Weiyang Sun wrote:
> I thought that the reading would have made a little more sense, and 
> been a bit more understandable, if section three had been put at the 
> beginning (example, the review of mathematical data types on page 8 
> and the definitions of surjective, bijective, etc.), because I tried 
> the tutorial problems after reading each section I thought was 
> associated with it.  Also, I was wondering what the ZFC in ZFC axioms 
> stand for. (pg 14, section 4.)
>
>   - Weiyang Sun
>
> -- 
> "The winner ain't the one with the fastest car, son. It's just the one 
> who refuses to lose."  --ralph earnhardt 



From sjcates@theory.csail.mit.edu Fri Feb 17 18:17:33 2006
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We won't go too much deeper on these topics in this class, but there's 
an entire class on logic an set theory (18.510) if you want more

Sonya

Eli B Stickgold wrote:
> I would most have liked to learn more about the section on Validity on page 7
> (the last paragraph).  I found it very interesting that we could have
> conjectures over any possible set and I'd like to hear some examples, because
> it seems that they would be rare, given how odd the sets we can choose are. 
> We've got {#t, #f}, the natural numbers, or just a set of strings.  It would be
> interesting to learn what kinds of logic apply equally well to all of these
> things.
>
> -Eli Stickgold
>   



From sjcates@theory.csail.mit.edu Fri Feb 17 18:22:27 2006
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Hi Robert,
Hopefully Russel's paradox is a little clearer after today's lecture.  
Don't worry about the typo.

Sonya

Robert A Rudd wrote:
> Here are my reading comments for the reading assignment. I also included
> below  the reason why they're late. I'm really sorry about this.
>
>
> Comments:
> The passage i'm choosing to cite is the short blurb about Russel's
> Paradox at the end of the reading(pg. 15 of 15). I found this to be the
> most difficult part of the reading. My main source of concern is I'm not
> sure why the problem with the proof is that W is not a set. The thing
> that seems most obvious to me is that you couldn't define "S not in S"(I
> think that's the correct term).I'm really not clear on why the statement
> "..we let S be W is invalid, because S ranges over sets, and W is not a
> set" is true.
>
>
>
>
> Reason why I was late:
>
> I accidentally sent it to the wrong address. I logged onto athena after
> class Thursday only to see:
>
> The original message was received at Wed, 15 Feb 2006 07:57:53 -0500
> (EST)
> from OUTGOING-AUTH.MIT.EDU [18.7.22.103]
>
>    ----- The following addresses had permanent fatal errors -----
> <042-probs@theory.lcs.MIT.EDU>
> I apparently forgot the 6 in 6042. Really sorry about that. I can
> forward the e-mail for authenticity if you'd like, however I understand
> if I have to take a penalty for this.
>
> Thanks,
> Robert Rudd
>   



From ubong@MIT.EDU Fri Feb 17 18:30:37 2006
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To: <6042-probs@theory.csail.mit.edu>
Subject: RE: Week 2 Comments
Date: Fri, 17 Feb 2006 18:30:25 -0500
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Thanks.

U.b.

-----Original Message-----
From: Sonya Cates [mailto:sjcates@theory.csail.mit.edu] 
Sent: Friday, February 17, 2006 5:34 PM
To: U.b.
Cc: 6042-probs@theory.csail.mit.edu
Subject: Re: Week 2 Comments


Hi U. b.,

This is an important distinction. I'm going to quote Professor Meyer on 
this one:

"validity applies to a single formula. Soundness refers to a deduction 
rule, which consists of a set of formulas (the antecedents that appear 
above the line) along with a single formula (the consequent that appears

below the line). The connection between sounds and validity is stated in

problem 2 at end of Notes 1. "

Sonya


U.b. wrote:
>
> On Validity and Soundness:
>
> These two concepts initially seemed somewhat equivalent to me. I guess

> I was a bit confused at first. However, upon revisiting past lectures,

> it turns out they are related but different; a proposition could be 
> sound but not necessarily valid. A valid proposition is always sound. 
> I guess I'm assuming soundness applies not just to propositional 
> logic/rules but to propositional formulas, and that there is a 
> difference between a rule and a formula. In any case, I'll probably 
> comb through the textbook to see more examples.
>
> Ubong (U.b.) Ukoh
>




From cak@mit.edu Sat Feb 18 20:21:08 2006
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> > pg 7 ln2 2.6 Validity:  ..., a formula now must evaluate to true no
> > matter WHAT values its...
> 
> I'm not sure what you're getting at here -- is there something in here
> that confused you, or are you pointing out a typo that I'm missing?

Was a typo. Fixed now. Thanks,
Christos.


From cak@mit.edu Sat Feb 18 21:04:28 2006
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> First, I believe that section 2.6 (Validity) is missing a "what" in the 
> second paragraph, right between "no matter" & "values". 

Typo fixed. Thanks!

> I've never before heard of the distinction between codomain and range
> (and I'm not sure I understand why the distinction is made)

In some cases the range and the codomain are equal, but in others they are
different. And, sometimes, it takes a lot of thinking to tell which case
we are in. Some practial aspects of this will be obvious when we get to
counting.

> nor seen Russell's paradox in that particular notation (since I am
> already somewhat familiar with it, I understand what's going on but it
> may be worthwhile to write it out more explicitly for people who've
> never seen it before).

I agree the paradox is written in a way that makes the thing pop out like
magic, and some people won't realize how it really works. However, you
have to admit there is some elegance to it. In any case, it is there 
mostly for culture, so that people roughly know what it is about. 

> Also, I don't believe it's mentioned before the next-to-last page that
> these are the Zermelo-Fraenkel Axioms, so at least spelling out the
> acronym in the last paragraph would be clearer.

Good point. 

> Lastly, I was surprised to hear that there are collections that are
> simply "too big" to be sets, especially because I don't have any clue
> what it _means_ to be too big to be a set. 

"Too big" is just a quick way to describe something a bit more subtle.  
Maybe a better phrase would be "too inclusive".

Then the phrase "too big to be a set" really means that
	this collection is so inclusive that, if we allow ourselves to 
	call it a set, the logical consequences of this are devastating 
        ---in the sense that they contain contradictions.

For example, the "set of all sets" is such a problematic set ---leading to
Russell's Paradox. You clearly see that this is too inclusive: so
inclusive, that it even contains itself (and that's how you get the
paradox).

It is also true that no finite set is problematic in this sense. No
countable set, either. Even huge sets (much greater than infinite
countable sets) are not problematic either. So, "too inclusive"  
collections start appearing beyond the biggest sets that we can actually 
construct. So, this justifies the "too big" expression.

Christos.




From cak@mit.edu Sat Feb 18 21:14:54 2006
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> I haven't worked much with sets before so I found the passage at the
> top of page 14 confusing; it would be great if you could explain the
> difference between f and f-hat, and the difference between the range
> and codomain.  Also, I don't think I really got Russell's Paradox on
> page 15.

This is a delayed response, and maybe you have clarified these issues
already. If not, then please let me know and we can discuss them --either
after lecture or in office hours.

They can all (difference of f/f-hat, difference of range/codomain,
Russell's paradox) be explained very easily on the whiteboard, with
drawings.  Really.

Christos.


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Actually, "too big" is technically correct: if there is an injection
between a collection and a set, then the collection is set.&nbsp; Since
injection is the definition of "size is &lt;="" for infinite sets, we
can say that anything that is the same or smaller size as a set, is
also a set.&nbsp; So a collection that is not a set must be "bigger" than
any set.<br>
regards, A.<br>
<pre class="moz-signature" cols="72">Albert R. Meyer
Hitachi America Professor of Engineering
MIT Computer Science &amp; AI Laboratory
The Stata Center, 32-G624
32 Vassar Street
Cambridge, MA 02139
617.253.6024
<a class="moz-txt-link-abbreviated" href="mailto:meyer@csail.mit.edu">meyer@csail.mit.edu</a>
<a class="moz-txt-link-freetext" href="http://theory.csail.mit.edu/~meyer/">http://theory.csail.mit.edu/~meyer/</a> </pre>
<br>
<br>
Christos A. Kapoutsis wrote:
<blockquote
 cite="midPine.LNX.4.44.0602182023070.1986-100000@parrot.csail.mit.edu"
 type="cite">
  <pre wrap="">
  </pre>
  <blockquote type="cite">
    <pre wrap="">First, I believe that section 2.6 (Validity) is missing a "what" in the 
second paragraph, right between "no matter" &amp; "values". 
    </pre>
  </blockquote>
  <pre wrap=""><!---->
Typo fixed. Thanks!

  </pre>
  <blockquote type="cite">
    <pre wrap="">I've never before heard of the distinction between codomain and range
(and I'm not sure I understand why the distinction is made)
    </pre>
  </blockquote>
  <pre wrap=""><!---->
In some cases the range and the codomain are equal, but in others they are
different. And, sometimes, it takes a lot of thinking to tell which case
we are in. Some practial aspects of this will be obvious when we get to
counting.

  </pre>
  <blockquote type="cite">
    <pre wrap="">nor seen Russell's paradox in that particular notation (since I am
already somewhat familiar with it, I understand what's going on but it
may be worthwhile to write it out more explicitly for people who've
never seen it before).
    </pre>
  </blockquote>
  <pre wrap=""><!---->
I agree the paradox is written in a way that makes the thing pop out like
magic, and some people won't realize how it really works. However, you
have to admit there is some elegance to it. In any case, it is there 
mostly for culture, so that people roughly know what it is about. 

  </pre>
  <blockquote type="cite">
    <pre wrap="">Also, I don't believe it's mentioned before the next-to-last page that
these are the Zermelo-Fraenkel Axioms, so at least spelling out the
acronym in the last paragraph would be clearer.
    </pre>
  </blockquote>
  <pre wrap=""><!---->
Good point. 

  </pre>
  <blockquote type="cite">
    <pre wrap="">Lastly, I was surprised to hear that there are collections that are
simply "too big" to be sets, especially because I don't have any clue
what it _means_ to be too big to be a set. 
    </pre>
  </blockquote>
  <pre wrap=""><!---->
"Too big" is just a quick way to describe something a bit more subtle.  
Maybe a better phrase would be "too inclusive".

Then the phrase "too big to be a set" really means that
	this collection is so inclusive that, if we allow ourselves to 
	call it a set, the logical consequences of this are devastating 
        ---in the sense that they contain contradictions.

For example, the "set of all sets" is such a problematic set ---leading to
Russell's Paradox. You clearly see that this is too inclusive: so
inclusive, that it even contains itself (and that's how you get the
paradox).

It is also true that no finite set is problematic in this sense. No
countable set, either. Even huge sets (much greater than infinite
countable sets) are not problematic either. So, "too inclusive"  
collections start appearing beyond the biggest sets that we can actually 
construct. So, this justifies the "too big" expression.

Christos.



  </pre>
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From meyer@imap.theory.csail.mit.edu  Tue Feb 21 02:17:12 2006
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Date: Tue, 21 Feb 2006 02:17:08 -0500
From: Soroush Vosoughi <soroush@MIT.EDU>
To: 6042-probs@theory.csail.MIT.EDU
Subject: Re: Reading assignment
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Hello,
Thank you very much for the explanation. It was very clear and easy to follow.
Thanks
-Soroush Vosoughi

Quoting Sonya Cates <sjcates@theory.csail.mit.edu>:

> Hi Soroush,
> One way of thinking about this would be to assign each element of A 
> either a 0 or a 1.  For any subset of A, 0 means that element is not 
> a member of the subset and 1 means that it is.  So there would be a 
> distinct binary string of length n for every possible subset of A.  
> There are 2^n binary strings of length n (do you know why this is 
> true?  if not ask me.) so there are also 2^n possible subsets of A.  
> Therefore the power set, which is defined as all possible subsets, 
> has 2^n elements.
>
> Here's an example:
> A={1,2,3}
> subset   string
> 1             100
> 2             010
> 3             001
> {1,2}      110
> {1,3}      101
> {2,3}      011
> {1,2,3}   111
> emptyset 000
>
>
> Sonya
>
> Soroush Vosoughi wrote:
>> Hello,
>> Passage 3.2.2 on page 10 has a defination of the power set. It says that the
>> power set is the collection of all subsets of a set A. It also says 
>> that if A
>> has n elements then P(A) would have 2^n elements. Why is that? I 
>> tried proving
>> it but couldn't. Could you please prove that in class.
>> Thank you
>> -Soroush Vosoughi
>>
>
>
>