What is wrong with the following argument that attempts, to show that if R is a relation...

What is wrong with the following argument that attempts, to show that if R is a relation on set S that is both symmetric and transitive, then R is also reflexive?

Since x R y implies y R x by the symmetric property, x R y and y R x imply x R x by the transitive property. Thus x R x is true for each so R is reflexive.