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"Let A be the set of all people in the world, R a relation on A defined by (a, b) ∈ R, if and only if a is a twin of b. Is this relation an equivalence relation? Explain!"

An equivalence relation is a reflexive, symmetric and transitive relation.

A = set of all people in the world R = {(a,b)| a and b are twins}

Okay so i thought this: This relation is not reflexive (a,a) or (b,b) because I am not my twin. But, it is symmetric because (a,b) and (b,a), that means I am my twin's twin and he/she is also my twin. Transitive in this case would basically mean reflexive, but like I said that is false.

Therefore my relation is not an equivalence relation...

Is my argument correct?

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Yes, your reasoning is correct.

BTW, with the alternative definition of $(a, b) ∈ R$ if and only if $a$ is a twin of $b$ or $a$ and $b$ are the same person, the relation $R$ becomes an equivalence relation.

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