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:
45380
Surjectivity implies injectivity and conversely
128305
$PGL(n, F)=PSL(n, F)$
42095
Existence in ZF of a set with countable power set
130998
Is $\exists x \in A ~:~ P(x)$ the same as $\exists x ~:~ x \in A \implies P(x)$
73454
Prove that $f$ has finite number of roots
41661
Show that the set of all finite subsets of $\mathbb{N}$ is countable.
39351
$f(X)$ is uncountable and hence $X$ is uncountable.
154559
For each $x\in X$ there exists a neighborhood of x s.t $A\cap U\neq \emptyset$ for only finitely many $A\in\mathcal{A}$. Show that f is continuous.
48278
Inequality about infinite cardinals
48615
Does the assertion that every two cardinalities are comparable imply the axiom of choice?
49592
Show that, if $f:A\to B$ is a function, with $A$ and $B$ being finite sets, and $|A|=|B|$, then $f$ is one to one iff $f$ is onto.
145873
Let $X$ and $Y$ be finite sets. Then the set $Y^X$ is finite and $\#(Y^X) = (\#Y)^{\#X}$.
98518
Cardinality of equivalence relations on N
120252
Equivalences of bounded sets definitions
111720
Is "have the same cardinality" a equivalence relation?
125683
Power set of a finite set is finite proof, no cardinality of power set
117912
How to show that $p$th moment being finite is equivalent to a limit existing