The question is, "Let R be the relation on the set of ordered pairs of positive integers such that $((a, b), (c, d)) ∈ R$ and only if $a+d=b+c$. Show that R is an equivalence relation."
There are two ways to prove this, but I only understand the second one.
The first way to proof: "By algebra, the given conditions is the same as the condition that $f((a,b))=f((c,d))$, where $f((x,y))=x-y$. Therefore, this is an equivalence relation."
I am not remotely sure of what they are doing…
Best Answer
You are given
$$a + d = b + c$$
as being equivalent to $(a, b) R (c, d)$ (i.e. $((a, b), (c, d))\in R$).
Then, you can do
$$a + d - b = c$$ (subtract b from both sides) $$a - b = c - d$$ (subtract d from both sides)
So then define $f((x, y)) = x - y$, and now you have the given statement. Now, it's easy to see the given relation is an equivalence relation, being that it now follows directly from the properties of equality.