Why isn’t reflexivity redundant in the definition of equivalence relation?

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

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