   Chapter 1.7, Problem 6TFE

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# Label each of the following statements as either true or false.Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric x R y implies y R x . Since R is transitive x R y and y R x implies x R x . Hence R is alsoreflexive and thus an equivalence relation on A .

To determine

Whether the statement, “Let R be a relation on nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies that yRx. Since R is transitive xRy and yRx implies that xRx. Hence R is reflexive and thus an equivalence relation on set A ” is true or false.

Explanation

Consider the statement, “Let R be a relation on nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies that yRx. Since R is transitive xRy and yRx implies that xRx. Hence R is reflexive and thus an equivalence relation on set A ”.

Counter example:

Let A={a,b,c}

Let R={(a,a),(a,b),(b,a),(b,b)} is relation on set A.

For (a,a)R implies that (a,a)R

For (a,b)R implies that (b,a)R

For (b,a)R implies that (a,b)R

For (b,b)R implies that (b,b)R

Thus the relation R is symmetric

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