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Another person who shares both mother and father with you is your full sibling. Is the relation "x S y meaning x is a full sibling of y" reflexive? Is it symmetric? Is it transitive?

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You share a mother and father with yourself, so the relation S is reflexive. If you share a mother and father with someone, then they share a mother and father with you, so S is symmetric. Finally, if you share a mother and father with someone, and they share a mother and father with a third person, then you also share a mother and father with that third person. Thus the relation S is transitive. This means that S defines an equivalence relation, partitioning people into sets of full siblings.

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More generally, if $f \colon A\to B$ is any function, then the relation $\sim_f$ defined by $$a\sim_f b \qquad \Leftrightarrow \qquad f(a)=f(b)$$ is an equivalence relation.

On ProofWiki they call this equivalence relation induced by $S$, I am not sure whether there are some other commonly used names. On ProofWiki you can also find a proof that it, indeed, is an equivalence relation.

In your case, if $P$ is the set of all people and $f \colon P \to P\times P$ is a function which assigns to any person the ordered pair (mother,father); your relation is precisely $\sim_f$.

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