This question is from the book Abstract Algebra: Theory and Applications by Thomas W. Judson.
Chapter 1 – Question 25
Determine whether or not the following relations are equivalence relations on the given set. If the relation is an equivalence relation, describe the partition given by it. If the relation is not an equivalence relation, state why it fails to be one.
(a) $x \sim y$ in $\mathbb{R}$ if $x \geq y$
(b) $m \sim n$ in $\mathbb{Z}$ if $mn > 0$
(c) $x \sim y$ in $\mathbb{R}$ if $|x – y| \leq 4$
(d) $m \sim n$ in $\mathbb{Z}$ if $m \equiv n \pmod{6}$