Visualizing a restriction function

Question

So I'm doing really basic review of functions to get back into the spirit of self-study. I came across this definition that I was hoping you guys could help me visualize:

$A \subset B, \ \ $

$f:A\to B,$defined by $f(a)=a, \ \ \forall a\in A,$

$ \\ g:B\to C ,$

then, $g|_{A}: A\to C$ is a restriction of g to A.

My question then is how would I go about visualizing this particular function? Below is how I see it working, but maybe someone has a more elegant picture?

Thanks guys!

Answer 1

I'm not too sure what's going on in your picture. There doesn't seem to be any mention of $g|_A$ and $g(B \setminus A)$ need not be disjoint from $g(A)$.

But anyway, we can think of $g|_A$ as $$g|_A = g(a) \, \, \quad \forall a \in A.$$ We don't need to care about how $g$ acts on $B\setminus A$. Really this restriction throws away all the information about how $g$ acts $B \setminus A$ and only keeps the information of how $g$ acts on $A$. We formalize the above view by defining $g|_A:A \to C$ by $g|_A = g(\mathrm{id}_A)$.

For the sake of visualisation, I'd draw something like this