TITLE: If $A(A(x)) = x$ for all $x$, then $A$ is one to one and onto
QUESTION [6 upvotes]: Suppose that $A$ is a mapping from a set $S$ to itself and $A(A(x)) = x$ for all $x \in S.$ Prove that $A$ is one-to-one and onto. 

Can someone please break this down, define one to one, and onto? I am new to this terminology.

REPLY [3 votes]: A function $f$ is called $1-1$ if $f(x) = f(y) \implies x = y$.
So suppose that
$$A(x) = A(y)$$
Applying $A$ to both sides, we see that
$$A(A(x)) = A(A(y))$$
But how can you write $A(A(x))$ and $A(A(y))$?

A function $g$ is called onto if for every $y$, there exists an $x$ for which $f(x) = y$. Given $y$, write
$$y = A(A(y))$$
So what input to $A$ gives $y$ as an output?