If $X$ is a finite set of cardinality $n$, where $n$ exists in $P$, show that the following conditions on a function $f: X \to X$ are equivalent:

Question

(a) $f$ is an injection (b) $f$ is a surjection (c) $f$ is a bijection

I know that (c) implies (a) and (b) and (a) and (b) imply (c). I also have the following definition that I've been playing around with:

If there exists a bijection from the set $X$ to the set $Y$, we write $ \#X=\#Y$ and we say that the sets $X$ and $Y$ have the same cardinality.

Since we are dealing with $f: X\to X$, does that imply that $f$ is an injection (since $f(x_1)=f(x_2)\to x_1=x_2$)

But I'm not sure what to make of this...