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

[AxiomOfChoice]

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

 

[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).

 

[AxiomOfChoice]

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).

Perfect. I think I understand now. Thank you.

 

[adriank]

You could also take the empty relation on a nonempty set.

 

[monguin61]

I think a good practical example of a relation with these properties is floating point equality - all floating point numbers equal themselves, but NaN != NaN, so the relation is not truly reflexive.