Equality is both symmetric and antisymmetric

"Equality is both symmetric and antisymmetric"

The statement "Equality is both symmetric and antisymmetric" is true, and it's a great example to understand these properties in the context of relations.

Let’s break it down:

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Equality as a Relation

Equality ( = ) is a relation defined on a set — say, the set of real numbers ℝ.

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Symmetric Relation

A relation $R$ is symmetric if:

If $a\, R\, b$, then $b\, R\, a$

For equality:

If $a = b$, then $b = a$

✅ This is clearly true, so equality is symmetric.

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Antisymmetric Relation

A relation $R$ is antisymmetric if:

If $a\, R\, b$ and $b\, R\, a$, then $a = b$

For equality:

If $a = b$ and $b = a$, then $a = b$

✅ This is also trivially true, because if both $a = b$ and $b = a$ hold, then obviously $a = b$

So equality is also antisymmetric.

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Conclusion

Equality is both symmetric and antisymmetric. This is possible because antisymmetry doesn’t prohibit symmetry — it just requires that if both directions hold, the elements must be equal, which they are in the case of equality.

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If you want a quick comparison:

Property	Does Equality Have It?	Why
Reflexive	✅ Yes	$a = a$ always holds
Symmetric	✅ Yes	$a = b \Rightarrow b = a$
Antisymmetric	✅ Yes	$a = b$ and $b = a \Rightarrow a = b$
Transitive	✅ Yes	$a = b$ and $b = c \Rightarrow a = c$

So equality is an example of a partial order and also an equivalence relation — in fact, it's the strongest form of relation: complete identity.