A bit confused about definition of set of mappings in Herstein's Algebra I have been just trying to start working with the books Topics in Algebra by I.N. Herstein, and I am having a bit of trouble understanding a definition.
It is,

Definition: If $S$ is a nonempty set then $A(S)$ is the set of all one-to-one mappings of $S$ onto itself.

I think my problem is that I am just having trouble thinking of an example of what this is representing? I mean lets say we were considering $S=\{1,2\}$ for example, then does A(s) contain some functions say, f, g for example for which $f:S \to S$ is bijective and g is as well?
My apologize if this is hard to understand my question, and it is possible my intuition is completely off, so I would really appreciate any explanations/insight about this. I am also curious if there is a more common name for this set $ A(S)$. 
Thanks all!
 A: Yes, by Herstein's definition, $A(X)=\{\text{bijective functions }f:X\to X\}$. 
Just for emphasis, since this seems to be something you're confused on, the elements of $A(X)$ are precisely the bijective functions from $X$ to $X$. There's nothing else in $A(X)$ other than that, and every such bijective function is included.
The operation of the group $A(X)$ is composition of functions:
$$\text{for any }f,g\in A(X),\;g\circ f\;\text{ is defined by the rule}\quad (g\circ f)(x)=g(f(x))$$
The more common notation (and name) for this set is $S_X$, the symmetric group on the set $X$. Here's the relevant Wikipedia page.
A: If $S=\{1,2\}$, then $A(S)$ consists of two functions $f,g$ where $f(1)=1,f(2)=2,g(1)=2,g(2)=1$. For other choices of $S$, construct $A(S)$ similarly as the set of all such functions. For instance, $A(\{1,2,3\})$ has six elements and $A(\mathbb{N})$ has uncountably many. A simple exercise to check your understanding would be to compute the number of elements in $A(\{1,2,...,n\})$.
