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### MAS111_09_quiz2

Course: MAS 111, Spring 2010
School: Nanyang Technological...
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111 MAS FOUNDATION OF MATHEMATICS QUIZ 2 DATE: 07/11/2009 1. Let A = {1, 2, 3, , 15}. (25 marks) (a) How many subsets of A contain all of the odd integers in A? (b) How many 10-element subsets of A contain exactly four odd integers? 2. On the set Z if all integers, dene the relation R by R = {(x, y ) Z Z : x2 y 2 is divisible by 5}. Prove that R is an equivalence relation. Find the equivalence classes of...

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111 MAS FOUNDATION OF MATHEMATICS QUIZ 2 DATE: 07/11/2009 1. Let A = {1, 2, 3, , 15}. (25 marks) (a) How many subsets of A contain all of the odd integers in A? (b) How many 10-element subsets of A contain exactly four odd integers? 2. On the set Z if all integers, dene the relation R by R = {(x, y ) Z Z : x2 y 2 is divisible by 5}. Prove that R is an equivalence relation. Find the equivalence classes of equivalence this relation on Z. 3. Let A = {1, 2, 3, 4, 5} and R be a relation on A dened by R = {(1, 1), (3, 3), (1, 3), (2, 3), (3, 2), (4, 2), (3, 5), (4, 3)}. (i) Is R symmetric? Justify your answer. (ii) Find R R1 . Is R R1 symmetric? (iii) Find the transitive closure of R. 4. Let D = {1, 2, 3, 4} and let be the subset relation on P (D). Draw the Hasse diagram of the partial order (P (D), ). (25 marks) 1
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