# A symmetric, transitive relation on a set that is not reflexive

1. Sep 5, 2010

### AxiomOfChoice

Can someone give an example of one? I can't think of one...

2. Sep 5, 2010

### Landau

Let X={a,b} (a and b distinct). Define the relation R on X by R={(a,a)}. Then R is symmetric and transitive, but not reflexive on X since (b,b) is not in R.

The point is that reflexivity involves a set ("reflexive on X": FOR ALL x in X we must have (x,x) in R), but symmetry and transivity are defined by means of an implication (IF ... is in R, THEN ... is in R).

3. Sep 6, 2010

### AxiomOfChoice

Perfect. I think I understand now. Thank you.

4. Sep 6, 2010