I am not exactly sure what these functions mean:
defined by for and
defined by for .
These two functions really are and right?
So if is a bijection with inverse , then if and only if so that . So this implies that sometimes?
This inverse image function is really useful.
I do not think you taken group theory, given the homomorphism then where is called the kernel it is extremely important concept.
Have you even taken linear algebra? Given a linear transformation then is defined exactly like above where is the zero vector of . This is another important concept.
The answer to the first question is: it is a foundational issue. That is, in a course on the foundation of mathematics, logic & sets, a function is defined as a relation with additional properties. If is a function from A to B then is also a relation from A to B. Now it is true then that is also a relation from A to B BUT it may not be a function.
However is always a function from to .