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

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Discussion Overview

The discussion revolves around identifying an example of a symmetric, transitive relation on a set that is not reflexive. Participants explore various examples and clarify the definitions of the properties involved.

Discussion Character

  • Exploratory, Conceptual clarification, Debate/contested

Main Points Raised

  • One participant requests an example of a symmetric, transitive relation that is not reflexive.
  • Another participant proposes the relation R={(a,a)} on the set X={a,b}, noting that it is symmetric and transitive but not reflexive since (b,b) is not included in R.
  • A similar response reiterates the same example and explanation, indicating that reflexivity requires all elements in the set to relate to themselves, while symmetry and transitivity are based on implications.
  • Another participant suggests that the empty relation on a nonempty set also fits the criteria of being symmetric and transitive but not reflexive.
  • One participant introduces the concept of floating point equality as a practical example, stating that while all floating point numbers equal themselves, NaN does not equal NaN, thus making the relation not truly reflexive.

Areas of Agreement / Disagreement

Participants present multiple examples and perspectives, indicating that there is no consensus on a single example, and various interpretations of reflexivity are discussed.

Contextual Notes

Some definitions and implications of reflexivity, symmetry, and transitivity may depend on specific interpretations or contexts, which remain unresolved in the discussion.

Who May Find This Useful

Readers interested in set theory, relations, and properties of mathematical structures may find this discussion relevant.

AxiomOfChoice
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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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