Finite equivalence class same cardinality For an equivalence relation $\sim$, if each partition has a finite number of elements, and $X$ is an infinite set, then is it true that $|X/\sim|=|X|$?
I can prove injectivity one way by defining the map $f:X/\sim\rightarrow X$ by picking an element from the equivalence class, but not sure how I would go about the other way.

The quotient set $X / {\sim}$ is a partition of $X$ into finite sets $[X]_i$ with $i < |X / {\sim}|$ (the axiom of choice here is implicit: $i$ is an ordinal less than the cardinal number $|X / {\sim}|$).
Map each $f_i : [X]_i \to \omega \times \{ i \}$ injectively since they are finite.
Take the union $\displaystyle \bigcup_{i < |X / {\sim}|} f_i : X \to \omega \times |X / {\sim}|$, which is injective, since the images of the component functions are disjoint.
So we have $$|X| \le |\omega \times X / {\sim}| \le |\omega \times X| = |X|$$ since $X$ is an infinite set.
$|\omega \times X / {\sim}| = | X / {\sim}|$ because $X / {\sim}$ is infinite (easily seen because otherwise, $X = \bigcup X / {\sim}$ is the finite union of finite sets).
