Chapter 9.1, Problem 51E

d the error in the “proof” of the following “theorem.”

“Theorem”:LetRbe a relation on a setAthat is symmetric and transitive. ThenRis reflexive.

Proof:Let $a\in A$.Take an element $b\in A$such that $\left(a,b\right)\in R$.BecauseR issymmetric, we also have $\left(b,a\right)\in R$. Now using the transitive property, we can conclude that $\left(a,a\right)\in R$because $\left(a,b\right)\in R$and $\left(b,a\right)\in R$.