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

In summary, the conversation discusses the concept of reflexivity, symmetry, and transitivity in relation to a given set and relation. The example given is that of a set X={a,b} and relation R={(a,a)}. It is concluded that R is symmetric and transitive, but not reflexive on X. The differences between these properties are explained, with reflexivity being defined by a set and symmetry and transitivity being defined by an implication. An example of a practical application of these properties is given as well.
  • #1
AxiomOfChoice
533
1
Can someone give an example of one? I can't think of one...
 
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  • #2
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
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.
 
  • #4
You could also take the empty relation on a nonempty set.
 
  • #5
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.
 

1. What is a symmetric, transitive relation?

A symmetric, transitive relation is a relationship between elements in a set where if element A is related to element B, then element B is also related to element A. Additionally, if element A is related to element B and element B is related to element C, then element A is also related to element C.

2. What does it mean for a relation to be reflexive?

A relation is reflexive if every element within the set is related to itself. In other words, for any element A in the set, A is related to itself.

3. Why is a symmetric, transitive relation not reflexive?

A symmetric, transitive relation is not reflexive because it does not satisfy the condition that every element must be related to itself. This means that there is at least one element in the set that is not related to itself.

4. Can you give an example of a symmetric, transitive relation that is not reflexive?

One example of a symmetric, transitive relation that is not reflexive is the "is parallel to" relation on a set of lines. If line A is parallel to line B, then line B is also parallel to line A. However, a line is not parallel to itself, so the relation is not reflexive.

5. What are some real-world applications of a symmetric, transitive relation that is not reflexive?

One real-world application is the "likes" relation on social media platforms. If person A likes person B's post, then person B is also liked by person A. However, a person cannot like their own post, so the relation is not reflexive. Another example is the "is similar to" relation in geometry. If triangle A is similar to triangle B, then triangle B is also similar to triangle A. However, a triangle is not similar to itself, so the relation is not reflexive.

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