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\tutorial{tutorial1}{Problems for Tutorial 1}{September 9, 1996}

\section {Warmup Problem}

\noindent Cast the following into propositional symbols, and verify or
refute:\\

{\em \noindent If Superman were able and willing to prevent evil, he would do
  so. If Superman were unable to prevent evil, he would be impotent.
  If he were unwilling to prevent evil, he would be malevolent.
  Superman does not prevent evil. If Superman exists, he is neither
  impotent nor malevolent. Therefore, Superman does not exist.}\\


\section {Using Peano's Axioms}

The following are Peano's axioms for defining the natural numbers
$N$.\\
\\
$\bullet  \quad 0 \in N$ \\
$\bullet  \quad \forall x \in N, x' \in N$ \\
$\bullet  \quad \forall x,y \in N, (x'=y') \Rightarrow (x=y)$\\
$\bullet  \quad \neg \exists x \in N$ such that $0 = x'$\\
$\bullet  \quad (Induction) \forall S \subseteq N$, if $0 \in S$ and $(x
\in S \Rightarrow x' \in S)$  then $ S = N$ \\

\noindent Now define addition as follows:\\
\\
$\bullet  \quad n + 0 = n$ \\
$\bullet  \quad n + k' = (n+k)'$ \\

\subsection{Additive Commutativity}

\noindent Use this to prove that addition is commutative, that is, that $\forall
m,n \in N, m+n = n+m$.\\ 

\subsection{Additive Associativity}

\noindent Now proceed to prove that addition is associative, that is, that $\forall
a,b,c \in N, (a+b)+c = a+(b+c)$. The previous result may be helpful here.\\

\subsection{Multiplication and the Distributive Law}

Define multiplication as follows:\\
\\
$\bullet  \quad a * 0 = 0$ \\
$\bullet  \quad a * k' = a*k+a$ \\

\noindent Now proceed to prove the following distributive law: $\forall
a,b,c \in N, a*(b+c) = a*b+a*c$.





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