Let be the relation on the set of ordered pairs of positive integers (i.e. ℤ+ × ℤ+) such that ((, ), (, )) ∈ is and only if + = + . Show that is an equivalence relation. Find the equivalence class of (1,2).
I did show that R is an equivalence relation. Next I tried to find the equivalence class like this: Let [x] be the equivalence class of (1,2). Therefore,
[x] = {(e, f) ∈ ℤ+ × ℤ+ | (e, f)R(1, 2)}
1 + f = 2 + e
From here onwards I don't understand what to do.
$\begin{align}[x]&=[(1,2)]\\&=\{(e,f)\in\Bbb Z^+\times\Bbb Z^+\mid(e,f)\mathrel{R}(1,2)\}&&\text{definition: equivalence class}\\&=\{(e,f)\in\Bbb Z^+\times\Bbb Z^+\mid e+2=f+1\}&&\text{definition: }R\\&=\{(e,f)\in\Bbb Z^+\times\Bbb Z^+\mid f=1+e\}\\&=\{(e,1+e)\mid e\in\Bbb Z^+\}\end{align}$
Please lemme know if that's clear enough.