# test2_key - COT3100 Spring2001 Test#2(Total 110 pts...

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COT3100 Spring’2001 Test #2 (Total 110 pts) Solutions 1. Consider the relation R = {( a , c ), ( b , d ), ( c , c ), ( d , a ), ( d , b )} on the set A ={ a , b , c , d }. a) (3pts) Find r ( R ), the reflexive closure of R . r ( R ) = R {( a , a ), ( b , b ), ( c , c ), ( d , d )}= = {( a , c ), ( b , d ), ( c , c ), ( d , a ), ( d , b ), ( a , a ), ( b , b ), ( d , d )} b) (3pts) Find s ( R ), the symmetric closure of R . s ( R ) = R R –1 = {( a , c ), ( b , d ), ( c , c ), ( d , a ), ( d , b ), ( c , a ), ( a , d )} c) (4pts) Find t ( R ), the transitive closure of R . One way is to find powers of R R 2 = {( a , c ), ( b , b ), ( b , a ), ( c , c ), ( d , c ), ( d , d )}; R 3 = {( a , c ), ( b , d ), ( b , c ), ( c , c ), ( d , a ), ( d , b ), ( d , c )}; R 4 = {( a , c ), ( b , b ), ( b , c ), ( c , c ), ( d , d ), ( d , c )} Or it is possible to take all pairs connected with a path: t ( R ) = {( a , c ), ( b , d ), ( b , b ), ( b , a ), ( b , c ), ( c , c ), ( d , a ), ( d , b ), ( d , d ), ( d , c )} 2. (8pts) Suppose R is a relation on a set A . Prove or disprove, that if R is symmetric and transitive, then R is reflexive. The proposition is false and can be disproved by the following example: A ={ x , y , z }, R ={( x , x ), ( x , y ), ( y , x ), ( y , y )} is symmetric and transitive, but is not reflexive, because ( z , z ) R . 3. Let R and S be relations on X . Determine whether each statement is true or false. If the statement is false, give a counterexample.

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test2_key - COT3100 Spring2001 Test#2(Total 110 pts...

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