I am notexactlysure 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?

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- Jul 9th 2007, 01:30 PMtukeywilliamsFunctions and Inverses
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? - Jul 9th 2007, 01:36 PMPlato
- Jul 9th 2007, 01:52 PMtukeywilliams
Yes, but why do we need these functions? Are they only useful for figuring out whether functions have inverses?

In other words, if then has no inverse? - Jul 9th 2007, 01:58 PMThePerfectHacker
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. - Jul 9th 2007, 01:59 PMtukeywilliams
Yes I have taken linear algebra.

- Jul 9th 2007, 02:01 PMThePerfectHacker
- Jul 9th 2007, 02:07 PMtukeywilliams
Yeah the kernel of a linear transformation is the set of vectors

**v**such that , i.e. the set of vectors that maps to the 0 vector.

So the inverse image function takes the kernel and maps it to the singleton set and vice versa. - Jul 9th 2007, 02:23 PMPlato
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 .