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# exam2prac - COT3100 Spring2001 Practice problems for the...

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COT3100 Spring’2001 Practice problems for the test#2 Relations. 1. Let A ={1, 2, 3}, B ={ w , x , y , z }, and C = {4, 5, 6}. Define the relation R A × B , S B × C , and T B × C , where R = {(1, w ), (3, w ), (2, x ), (1, y )}, S ={( w , 5), ( x , 6), ( y , 4), ( y , 6)}, and T ={( w , 4), ( w , 5), ( y , 5)}. a) Determine R ( S T ) and ( R S ) ( R T ). b) Determine R ( S T ) and ( R S ) ( R T ). 2. Consider the relation T over real numbers: T = {( a , b ) | a 2 + b 2 =1}. Define all properties of this relation (reflexive, irreflexive, symmetric, anti-symmetric, and transitive). 3. Let R A × A be a symmetric relation. Prove that t ( R ) is symmetric. 4. Prove or disprove the following propositions about arbitrary relations on A : a) R 1 , R 2 symmetric R 1 R 2 is symmetric. b) R 1 , R 2 transitive R 1 R 2 is transitive. 5. Prove or disprove that for any relation R A × A ts ( R ) st ( R ). 6. Let R be a symmetric and transitive relation on set S . Furthermore, suppose, that for every x S there is an element y S such that xRy . Prove that R is equivalence relation.

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exam2prac - COT3100 Spring2001 Practice problems for the...

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