sampleexam1 - Determine whether the function f from the set...

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COT3100 Sample Midterm 1 Question 1 (10pts) Show that ( p q ) ( ¬ p r ) ( q r ) is a tautology. Question 2 (10pts) Prove that ¬ ( p ( ¬ p q )) ≡ ¬ p ∧ ¬ q by showing a sequence of logical equivalences. Question 3 (10pts) Determine the truth value of the following statements if the domain of each variable consists of all real numbers. (a) x y ( x 2 = y ) (b) x y ( xy = 0) Question 4 (10pts) Express the negation of the following statements so that all negation symbols immediately precede predicates. (a) x yP ( x, y ) ∨ ∀ x yQ ( x, y ) (b) x y ( P ( x, y ) Q ( x, y )) Question 5 (10pts) Prove that if n is an odd integer, then n 2 is odd. Question 6 (10pts) Find the power set and the cardinality of each of these sets. (a) { a, b } (b) {∅ , {∅}} 1
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Question 7 (10pts) Prove that (a) A B = A B ; (b) A ( B - A ) = A B Question 8 (10pts)
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Unformatted text preview: Determine whether the function f from the set of real numbers to itself is a bijection. If f ( x ) is a bijection, please nd f-1 ( x ) . (a) f ( x ) = x + 2 (b) f ( x ) = 3 x 2-1 Question 9 (10pts) For the following lists of integers, provide a simple formula or rule that gener-ates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. (a) 3 , 6 , 11 , 18 , 27 , 38 , 51 , 66 , 83 , 102 , . . . (b) 7 , 11 , 15 , 19 , 23 , 27 , 31 , 35 , 39 , 43 , . . . Question 10 (10pts) Prove that between every two rational numbers there is an irrational number. 2...
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sampleexam1 - Determine whether the function f from the set...

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