I have the natural numbers $\mathbb{N}$, in ZF.
I want to construct the integers $\mathbb{Z}$ taking the quotient respect to usual the equivalence relation $R$ on $\mathbb{N}\times\mathbb{N}$ that is $(a,b)R(c,d) \iff a+d=b+c$.
I have to construct a set $X=\mathbb{Z}$ that contains the equivalence classes.
What is the correct ZF formulation for this set?
I think it's something like $X:=\{a \ | (\ b\in a, c\in a) \Rightarrow (b,c)\in R \}$
The problem is: if $a$ are the particular equivalence classes, from what set should I take them (for let $X$ be a ZF set)?