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Test 2 Solutions

# Test 2 Solutions - F06/CS 191 Suggested Solutions to TEST...

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F06/CS 191 : Suggested Solutions to TEST TWO Discrete Structures I 11/2/2006 OPEN BOOK Name_____________________ There are 5 problems with 20 points each. 1 Let X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}. Define the relation R on X as follows. ( a , b ) R if a mod 3 = b mod 3 for any a , b X (a) Prove that R is an equivalent relation. Proof. We need to show that R is reflexive, symmetric and transitive. (i) Because a mod 3 = a mod 3 for any number a , we have ( a , a ) R . So, R is reflexive. (ii) If ( a , b ) R , then we have a mod 3 = b mod 3. This implies b mod 3 = a mod 3. Therefore, ( b , a ) R. So R is symmetric. (iii) If ( a , b ) R and ( b , c ) R , then we have a mod 3 = b mod 3, and b mod 3 = c mod 3, which implies that a mod 3 = c mod 3. Therefore, we have ( a , c ) R . The transitivity is also proved. Therefore, R is an equivalence relation. (b) List the elements of [3], the equivalent class containing 3. [3] = {3, 6, 9, 12, 15} 1

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2 Let X = {2, 3, 4, 5}, Y = {6, 9, 10, 12}, Z = {2, 7, 11, 13} be three sets. We define two relations.
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Test 2 Solutions - F06/CS 191 Suggested Solutions to TEST...

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