Let $X = \Bbb N \times \Bbb N$ be the set of pairs of positive integers. Consider the equivalence relation $(a, b) \sim (c, d)$, if $a + d = c + b$.
Prove that there is a bijection $F: X{/} \to \Bbb Z$, where $X{/}\sim$ is the set of equivalence classes.
I tried defining a function $g: X \to \Bbb Z$ such that $g(a,b) = a - b$ and showed this is surjective. But it fails infectivity test when changing the domain from $X$ to $X{/}\sim$. Could someone please help me define a bijective function between $X{/}\sim$ and $\Bbb Z$?
The problem asks us to prove the existence of a bijection, which is purely dependent on the cardinality structure, i.e. it suffices to show $X/\sim$ is countably infinite.
In fact, $X/\sim$ is infinite, since $(1,n)\not\sim (1,m)$ whenever $m\ne n$, and therefore $X/\sim$ contains an infinite subset $\{[(1,n)]: n\in\mathbb{Z}_+\}$, where $[\cdot]$ is the equivalence class of $\cdot$.
On the other hand, $X/\sim$ is at most countable, the reason is as follows: For each equivalence class $A$, select one element $x_A$ in the equivalence class. This gives an injection $f:X/\sim\to X$, $A\mapsto x_A$. Since $X$ is countable, we infer so $X/\sim$ is at most countable.
Therefore $X/\sim$ is countably infinite.