In my Abstract Algebra class, the professor defined a restriction as
Given $ X\xrightarrow{f} Y $ and a non-void subset $S$ of $X$ define $ f \mid S\xrightarrow{S} Y $ by $(f \mid S )(s) = f(s), \forall s \in S$
He literally just pulled this one out of a hat without explaining what he meant by restriction or even what was meant by the definition.
If anyone could enlighten me, I'd much appreciate it.
On
The restriction of a function $f:X\to Y$ to any subset of $S\subseteq X$ (empty or not) simply refers to the creation of a new function $g:S\to Y$ which is defined as follows: For all $s\in S$: $g(s)=f(s)$. So, the formula (if you like) is unchanged, but the only thing that changes is that the set of inputs for the function was restricted. It is common to denote $g:S\to Y$ by $f\mid _S$.
Restricted here means much the same as it does in ordinary English. For example, think of the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^2$.
This function has some defects. For one thing, it is not one to one, since $f(-3)=f(3)$.
We can define a new function $g$, from the set $\mathbb{R}^+$ of non-negative integers to $\mathbb{R}$, by $g(x)=x^2$. This function is $f$ restricted to $\mathbb{R}^+$. Note that $g$ is one to one.
In the notation used in your course, $g$ is $f|\mathbb{R}^+$.
The only "inputs" that $g$ accepts are elements of $\mathbb{R}^+$, but it does the same thing to them as $f$ did.