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x1-sample - M42 Discrete Mathematics Spring 2011 M Stanley...

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M42 — Discrete Mathematics Spring 2011 M. Stanley Sample First Exam Do not use any aids (calculators, computers, notes, etc.) on this exam. When asked to provide proofs, be sure to give careful, detailed and complete explanations in full English sentences. Form is as important as content. (1) ( 1 point each ) Let A = { 1 , 2 } and B = { 2 , 3 } . Let the universal set U = { 1 , 2 , 3 , 4 } . Find (a) A B = { 1 , 2 , 3 } (b) A B = { 2 } (c) A B = { 1 , 3 } (d) A × B = { (1 , 2) , (1 , 3) , (2 , 2) , (2 , 3) } (e) ( A ) = {∅ , { 1 } , { 2 } , { 1 , 2 }} (f) A \ B = { 1 } = { 2 , 3 , 4 } (2) ( 3 points ) Use a truth table to show that (( p ∨ ¬ q ) q ) q . Answer: p q ¬ q p ∨ ¬ q ( p ∨ ¬ q ) q T T F T T T F T T F F T F F T F F T T F Because the pattern of T ’s and F ’s in the q column and the ( p ∨ ¬ q ) q column match, these two formulas are logically equivalent. (3) ( 3 points ) If possible, find an assignment of T and F to the propositional variables p , q , r , and s such that ( p q ) ( r s ) is T , but ( p r ) ( q s ) is F . p F q T r F s F
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Sample First Exam (4) ( 1 point each ) Suppose that the universe of discourse is { a, b, c } and that
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