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University of Ontario Institute of Technology *

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2110U

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Computer Science

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Nov 24, 2024

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pdf

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2

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MATH 2080U / CSCI 2110 Term Test 1 Page 2 of 5 1. (a) [5 marks] Prove that the propositions ( p ∨ ¬ q ) → ( ¬ p ∧ ¬ q ) and ¬ p are logically equiv- alent using logical identities . (b) [5 marks] Prove that the premises ∃ x ¬ Q ( x ) , ∀ x ( P ( x ) → Q ( x )) , and ∀ x ( P ( x ) ∨ R ( x )) lead to the conclusion ∃ xR ( x ) .
MATH 2080U / CSCI 2110 Term Test 1 Page 3 of 5 2. (a) [5 marks] Prove using ONLY the definitions of even and odd integers that n is an odd integer if and only if ( n + 1 ) 2 + 6 is an even integer. (b) [5 marks] Prove that if 1 < a < 2 and b = 1 + √ a - 1 then 1 < b and b < a . HINTS: 1. For any real number x , if 0 < x < 1, then x < √ x and if 1 < x then √ x < x . For examples: 0.81 < √ 0.81 = 0.9 and 1.1 = √ 1.21 < 1.21. 2. Start by isolating for the square root in the second premise.
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