Let R be the relation on $Z × Z$, that is elements of this relation are pairs of pairs of integers, such that $((a, b),(c, d))\in R$ if and only if $a + d = b + c$. Show that R is an equivalence relation.
I'm not sure how to explain this but I have some ideas:
1st Idea:
If R is a relation in which Z = {all integers} the R is an equivalent relation since it is reflexive, symmetric and transitive. So if a new relation R is a relation ((a, b),(c, d)) ∈ Z × Z, the does that mean that the new relation is also and equivalent relation because the original relation is equivalent?
2nd Idea:
So if the relation is reflexive then $a + d = a + d$. Since any integer is equal to itself, this is true. If it is symmetric then $b + c = a + d$. This is true because if $x = y$, then $y = x$. And it is transitive because if we add $(e,f)$ it the relation would still hold.
Hint: $R$ is a relation on the set $\mathbb{Z}\times\mathbb{Z}$. In order to show that $R$ is reflexive, you must show that if $q\in \mathbb{Z}\times\mathbb{Z}$ then $q\sim q$; that is, that for every $(a,b)\in\mathbb{Z}\times\mathbb{Z}$ that $(a,b)\sim (a,b)$. The other properties of an equivalence relation are similar. Does that help?