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mid-term-2009-make-up - B ∪ C A ∪ B ∩ C and A ∩ B...

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Nanyang Technological University Mid-Term Test 2009 — 2 MAS 111 Foundations of Mathematics Time Allowed: 1 Hour Instructions to the candidates The examination consists of FIVE (5) problems, each of which is of 20 marks. Answer all questions. The marks for each question are indicated at the end of each question. All calculations and answers should be accompanied by appropriate explanations.
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Problem 1 Prove that log 3 7 is irrational. Problem 2 Prove that the argument form with premises ( p t ) ( r s ) , q ( u t ) , u p and ¬ s and conclusion q r is valid. Problem 3 Let A = { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , B = { 3 , 4 , 7 , 8 , 9 , 10 } and C = { 2 , 4 , 6 , 8 , 10 } and U = { 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 } . List the elements in A - C , ( A
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Unformatted text preview: B ) ∪ C , A ∪ ( B ∩ C ) and A ∩ B , where X is the complement of X relative to U . Problem 4 (a) Let A be a set of size n . Prove that the power set P ( A ) of A has size 2 n . (b) Let B and C be sets with | C | = 2 . If there are 128 distinct functions from B to C , what is | B | ? Justify your answer. (c) Let D be a set of size n with n ≥ 2 . How many distinct reflexive and symmetric relations are there on D ? Justify your answer. Problem 5 Prove that if k is odd, then 2 n +2 divides k 2 n-1 for all positive integers n ....
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