Question

What is wrong with the following argument? Note the argument supposedly shows that any symmetric and transitive relation R on A must also be reflexive.
Let R be a relation in A × A that is symmetric and transitive. Using symmetry, if (x,y) ∈ R then (y,x) ∈ R. Hence, both (x,y) and (y,x) are in R. Since (x,y) and (y,x) ∈ R, by transitivity, we have (x,x) ∈ R. Therefore, R is reflexive.