Testing for the equivalence relation

[Quoted] [Quoted] I'm seeking some help on a rather basic and trivial question.

Is there a case where a relation's extension proves to be symmetric and transitive, but the relation is not an equivalence relation? In other words, can a relation be symmetric and transitive, but not be reflexive?

If I recall correctly, by definition, a relation is an equivalence relation if, and only if, the relation is reflexive, symmetric, and transitive. It seems intuitive that if a relation has the properties of being symmetric and transitive, the relation must also be reflexive.

My current rationale (that I want disproved) is the following:

Hypothesis: If a binary relation R is symmetric and transitive, it is an equivalence relation (ie. it implies reflexivity - in contrast to the classic definition).

By definition of symmetric property of a relation,

   for all x,y that are elements of A, x R y --> y R x.

By definition of transitive property of a relation,

   for all x, y, z that are elements of A, x R y and y R z --> x R z.

Reflexivity of the relation is implied because

  If x R y and y R x, then x R x (by the definition of transitivity), and therefore R is reflexive.

Thus, if relation R is symmetric and transitive, it is an equivalence relation.

I'm looking for a counterexample to this.

Thanks,

• Dan