Problem_Set_01

# Problem_Set_01 - %&amp;LaTeX \ \documentclasscfw_article \

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\documentclass{article} \ \newcommand{\tab}{\makebox[4em]{}} \ \begin{document} \ \begin{center} \textbf{ArsDigita University} \ \end{center} \ \begin{center} \textbf{Month 2: Discrete Mathematics - Professor Shai Simonson} \ \end{center} \ \begin{center} \textbf{Problem Set 1 -- Logic, Proofs, and Mathematical Reasoning} \ \end{center} \ \begin{enumerate} \item { Logic Proofs } \begin{enumerate} \item Prove that $a \rightarrow b$ is equivalent to $\neg b \rightarrow \neg a$ using a truth table. \\ \item Prove it using {\em algebraic} identities. \\ \item Prove that $a \rightarrow b$ is not equivalent to $b \rightarrow a$. \end{enumerate} \item {Aristotle's Proof that the Square Root of Two is Irrational.}\\ \begin{enumerate} \item Prove the {\em lemma,} used by Aristotle in his proof, which says that if $n^{2}$ is even, so is $n$. (Hint: Remember that $a \rightarrow b$ is equivalent to $b \rightarrow \neg a$.\\ \item Prove that the square root of 3 is irrational using Aristotle's techniques. Make sure to prove the appropriate lemma.\\ \item If we use Aristotle's technique to {\em prove} the untrue assertion that the square root of 4 is irrational, where {\em exactly} is the hole in the proof?\\ \item Using the fact that the square root of two is irrational, prove that $\sin (\frac{\pi}{4})$ is irrational.

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## This document was uploaded on 10/01/2011.

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Problem_Set_01 - %&amp;LaTeX \ \documentclasscfw_article \

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