A question about cardinal number.

Question

Let $X$ be a infinite set and $n$ be a positive integer. We denote the cardinal number of $X$ by $|X|$ and denote the family of all subsets of X which contains n elements by $\mathfrak{F}$. Then $|\mathfrak{F}|=|X|$.

By the fact that $|X^{n}|=|X|$, I have proved that $|\mathfrak{F}|\leq |X|$ and the problem is to construct a injection from $X$ to $\mathfrak{F}$. Anyway, any help is appreciated. Thanks!

Answer 1

Fix a set $A$ of $n-1$ elements of $X$; the map $X\setminus A\to\mathfrak{F}:x\mapsto A\cup\{x\}$ is an injection. Now use (or prove) the fact that $|X|=|X\setminus A|$.

Answer 2

Since we already use the axiom of choice, here's an easy way to construct a surjection from $\mathfrak F$ onto $X$.

Fix a linear ordering on $X$ without a maximal element, and map $A\in\mathfrak F$ to its least element according to that order. I will leave it to you to show this is a surjection.

Answer 3

Another way to prove this using AC (that you are already using to prove the cardinal equality from the inequalities), is to well-order $X$ with a limit ordinal, and then map each element $x\in X$ to the set of the first $n$ elements of $X$ starting at $x$.

Actually this injection is an inverse of the surjection suggested by @AsafKaraglia above, but I find the constructive injection more intuitive.