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