MAS111_09_quiz2 - A defined by R = { (1 , 1) , (3 , 3) ,...

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MAS 111 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 inte- gers? 2. On the set Z if all integers, define the relation R by (25 marks) R = { ( x,y ) Z × Z : x 2 - y 2 is divisible by 5 } . Prove that R is an equivalence relation. Find the equivalence classes of this equivalence relation on Z . 3. Let A = { 1 , 2 , 3 , 4 , 5 } and R be a relation on
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Unformatted text preview: A defined 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 ∩ R-1 . Is R ∩ R-1 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 ) , ⊆ ). 1...
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This note was uploaded on 01/23/2011 for the course MAS 111 taught by Professor Drchansongheng during the Spring '10 term at Nanyang Technological University.

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