Problem_Set_02

# Problem_Set_02 - %&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 2 -- Sets, Functions, Big-O, Rates of Growth} \ \end{center} \ 1.\tab Prove by formal logic:\\ a.\tab The complement of the union of two sets equals the intersection of the complements.\\ b.\tab The complement of the intersection of two sets equals the union of the complements.\\ c.\tab (\textit{B} \ensuremath{-}\textit{A}) \ensuremath{\cup} (\textit{C} \ensuremath{-} \textit{A}) = (\textit{B} \ensuremath{\cup} \textit{C}) -- \textit{A}. \\ d.\tab If two sets are subsets of each other then they are equal. d

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2.\tab Generalize De Morgan's laws for \textit{n} sets and prove the laws by induction. b 3.\tab Prove by induction on the size of the set, that the power set \textit{P(A)} has cardinality \textit{2}$^{\mathit{{\textbar}A}}$$^{{\textbar}}$. c \textit{4.\tab } \textit{A} \ensuremath{\oplus} \textit{B} is defined to be the set of all elements in \textit{A} or \textit{B} but not in both \textit{A} and \textit{B.} \\ a.\tab Determine whether or not \ensuremath{\oplus} is commutative. Prove your answer. \\ b.\tab Determine whether \ensuremath{\oplus} is associative. Prove your answer.\\ c.\tab Determine whether \ensuremath{\oplus} can be distributed over union. Prove your answer.\\ d.\tab Determine whether \ensuremath{\oplus} can be distributed over intersection. Prove your answer. P 5.\tab Assume a universal set of 8 elements. Given a set \textit{A = a}\$_{\mathit{7}}
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## This document was uploaded on 10/01/2011.

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Problem_Set_02 - %&amp;LaTeX \documentclasscfw_article

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