What is the flaw in the following argument which supposedly

What is the flaw in the following argument, which supposedly shows that any relation R on X that is symmetric and transitive is therefore reflexive? Let x X. Using symmetry we have that (x, y), (y, x) R and thus by transitivity (letting x take the role as both x and z) we have that (x, x) R. Therefore R is reflexive.

Solution

The every ordered pair (a,a) must belong to R, for the function to be a reflexive relation

This is the loop whole in the proof, since let us suppose a set is {1,2,3,4} and the relation defined is

R = { (1,2),(2,1),(1,1) }

The relation is both symmetric and transitive but not reflexive, since other (a,a) entries i.e. (2,2), (3,3) and (4,4) are not present in the relation set