An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity.

Doesn’t symmetry and transitivity implies reflexivity? Consider the following argument.

For any a and b,

aRb implies bRa by symmetry. Using transitivity, we have aRa.Source: Exercise 8.46, P195 of

Mathematical Proofs, 2nd (not 3rd) ed. by Chartrand et al

**Answer**

Actually, without the reflexivity condition, the empty relation would count as an equivalence relation, which is non-ideal.

Your argument used the hypothesis that for each a, there exists b such that aRb holds. If this is true, then symmetry and transitivity imply reflexivity, but this is not true in general.

**Attribution***Source : Link , Question Author : Chao Xu , Answer Author : Akhil Mathew*