Equivalence relation $(a,b) R (c,d) \Leftrightarrow a + d = b + c$ Suppose $A$ is the set composed of all ordered pairs of positive integers. Let $R$ be the relation defined on $A$ where $(a,b) R (c,d)$ means that $a + d = b + c$.
(a) Prove that $R$ is an equivalence relation.
Here is what I have so far. Is this correct?
Reflexive:
$a \sim a$
$\implies$ $a+b=a+b$; $(a,b) R (c,d)$
Symmetric:
if $a \sim b$ then $b \sim a$
$\implies$ if $a+d=b+c$ then $c+b=d+a$
Transitive:
if $a \sim b$ and $b \sim c$ then $a \sim c$
$\implies$ if $a+d=b+c$ and $c+f=d+e$ then $a+d=d+e$
(b) Find $[(1,1)]$.
I'm not sure how to approach this.

Your proof regarding the fact that $R$ is in fact an equivalence relation is indeed correct.
Now think what the equivalence class of $(1,1)$ could be, since $(1,1) R (1,1)$ you know that everyone in the equivalence class has the property that $(a,b)$ holds $a+1=b+1$, therefore $a=b$.
This means that the equivalence class of $(1,1)$ is $(a,a)$ for $a\in\mathbb{N}$.
