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.

HINT: The equivalence class of $(1,1)$ is made up of all pairs $(x,y)\sim(1,1)$. Write explicitely what the latter means and get a relation that need to be satisfied by $x$ and $y$.
