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!
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.