Equivalence relation - reflexity

Question

I need to prove that relation is an equivalence relation. Equivalence relation means it satisfies reflexity, symmetry, and transitivity.

If I was given an set of numbers S=(-1,1) and for example for -(1/2) relation is not reflexive, but for 1/2 it is reflexive. Does that mean that in the end relation is not reflexive, because it's not reflexive for all numbers from that set?

Thank you!

Reflexivity means that $aRa$ for all $a\in S$. – Angina Seng Nov 23 '18 at 07:02 — Angina Seng, Nov 23 '18 at 07:02
So, in this case it's not reflexive. Thank you! – Haus Nov 24 '18 at 17:20 — Haus, Nov 24 '18 at 17:20

score 0 · Answer 1 · answered Jun 26 '20 at 05:48

As noted in the comments, reflexivity has to hold for all elements of the set of concern. More precisely, let $R$ be a relation on $S$. Then, for all $s \in S$, we must have $(s,s) \in R$ (or $sRs$ depending on your notation of preference). We might write this as $\forall s \in S, (s,s) \in R$.

It only takes a single counterexample to bring the entire thing toppling down as a result. It has to hold for literally everything in $S$ or it's not reflexive.

Equivalence relation - reflexity

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