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\pset{2}{February 16, 1995}{In recitation, February 24}

%	{\bf Reading:\ }


\bigskip
\begin{problems}

% Problem 1
\problem 
Define the composition $P\circ Q$ of two relations as follows: \\
If $Q \subseteq A\times B$ and $P \subseteq B \times C$ then $P \circ Q =
\{(a,c) : 
%\mbox{ there exists } b \mbox{ such that } (a,b) \in P \mbox{ and
\exists b \mbox{ such that } (a,b) \in Q \mbox{ and } (b,c) \in P\}$.  
Note that $P \circ Q \subseteq A \times C$.  

Let $R \subseteq S \times S$ be a relation.  
Prove or disprove each of the following statements.
\bparts
\ppart If $R$ is transitive, then $R \circ R$ is transitive.
\ppart If $R$ is transitive, then $R \circ R \subseteq R$.
\ppart If $R \circ R \subseteq R$, then $R$ is transitive.
\eparts

% Problem 2
\problem  Prove or disprove the following identities:
\bparts
\ppart $A \times (B \cap C) = (A \times B) \cap (A \times C)$
\ppart $(A \cup B) \times (C \cup D) = (A \times C) \cup (B \times D)$
\ppart $(A - B) \times (C - D) = (A \times C) - (B \times D)$
\ppart $(A \bigtriangleup B) \times (C \bigtriangleup D) = (A \times C) \bigtriangleup (B \times D)$
\eparts

% Problem 3
\problem  
Let $f^{(i)}(x)$ denote the $i$th derivative of $f(x)$.
Prove that if $f(x)$ and $g(x)$ are real-valued functions of one
variable with derivatives of all orders, then
\[
(f(x)g(x))^{(n)} = \sum_{j=0}^{n} {n \choose j}f^{(n-j)}(x)g^{(j)}(x)
\]

% Problem 4

\problem 
You are given an unlimited supply of $4$-cent and $7$-cent
stamps.  What is the largest amount of postage  that you cannot make
with these two denominations?  Prove your answer. 
%	Come up with a conjecture and prove it.

\end{problems}

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