Consider the following relation on $\Bbb R$:
$$xRy \iff |x - y| \in [-1,1]$$
I have to find out whether it is an equivalence relation, for which it must be symmetric, reflexive, and transitive.
It is indeed symmetric, since $xRy$ implies $yRx$, for instance
$$2R1 \iff |2 - 1| = 1 \iff |1 - 2|=1 \iff 1R2$$
I know it is not a transitive relation however, considering $x = 0, y = 1$, and $z = 2$. And although I now know his means it can never be an equivalence relation, I want to find out how I can check if the relation is reflexive, $xRx$. How do I do so?
We have $xRx$ if and only if $|x - x| \in [-1, 1]$ if and only if $0 \in [-1, 1]$, which is true for all $x \in \mathbb{R}$, so the relation is indeed reflexive.