Problem_Set_01

Problem_Set_01 - %&amp;LaTeX \ \documentclasscfw_article \

This preview shows pages 1–2. Sign up to view the full content.

\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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This document was uploaded on 10/01/2011.

Page1 / 4

Problem_Set_01 - %&amp;LaTeX \ \documentclasscfw_article \

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online