rec0614_key - COT3100 Summer2001 Recitation on relations...

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COT3100 Summer’2001 Recitation on relations (Solutions) 06/14/2001 1. Let A ={1, 2, 3}, B ={1, 2, 3, 4}, and R 1 ={(1,2), (2,2), (3,3)} and R 2 = {(1,2), (1, 3), (1,4)} are relations from A to B . Find R 1 R 2 , R 1 R 2 , R 1 - R 2 , R 2 - R 1 . R 1 R 2 = {(1, 2), (2, 2), (1, 3), (1, 4), (3, 3)} R 1 R 2 = {(1, 2)} R 1 - R 2 = {(2, 2), (3, 3)} R 2 - R 1 = {(1, 3), (1, 4)} 2. Given the set A = {2, 3, 4, 8, 9, 12, 18}, define a relation T over A such that T = {( a , b )| a A and b A and ab is a square number, i.e., ab = c 2 for some integer c }. Answer the following two questions. a) Use a directed graph to depict the relation T defined above. b) Determine if relation T satisfies each of the properties: irreflexive, symmetric, and transitive. Not irreflexive (it is reflexive) symmetric transitive 3. Suppose R and S are symmetric binary relations on a set A. Must the following relations be symmetric? Give either proofs or counterexamples to justify your answers. a) R S Proof. Assume ( x , y ) is arbitrary element from R S , i. e. ( x , y ) R S . We need to prove that ( y , x ) R S given R and S are symmetric. If ( x , y ) R S , then ( x , y )
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rec0614_key - COT3100 Summer2001 Recitation on relations...

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