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

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An example of a symmetric and transitive relation that is not reflexive is the relation R defined on the set X={a,b} as R={(a,a)}. This relation is symmetric and transitive because it satisfies the implications required for those properties, but it fails to be reflexive since (b,b) is not included in R. The discussion clarifies that reflexivity requires every element in the set to relate to itself, while symmetry and transitivity are based on conditional implications. Additionally, the empty relation on a nonempty set is also cited as an example of such a relation. Floating point equality is mentioned as a practical example, where NaN is not equal to itself, highlighting the non-reflexive nature of the relation.
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Can someone give an example of one? I can't think of one...
 
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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).
 
Landau said:
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.
 
You could also take the empty relation on a nonempty set.
 
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.
 

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