Can the inverse of set-valued function also be a set-valued function?

Question

Can the inverse of set-valued function also be a set-valued function? If so, what we call the set-valued function whose inverse is a single-valued function?

Thanks very much!

Answer 1

Yes. A set-valued function is a function whose values are sets. The inverse of a set-valued function has to have an input that is a set. A function whose input is a set is called a set function. Therefore a set-valued function has a set-valued inverse iff the function is a set-valued set function. The inverse of such a function is also a set-valued set function.

You could say, the set-valued function whose inverse is a single-valued function is a single-value-argumented set-valued function, but you should declare that.

Answer 2

Yes. The definition of a set-valued, or multi-valued function is such that it is essentially a function in which uniqueness of images is not necessary:

A set-valued function from A to B is a binary relation R from A to B such that for each element a in A, there is at least one b in B so that aRb.

By this definition, any function from a set A to a set B is trivially a set valued function from A to B:

In particular, take any bijection from a set A to a set B. It is therefore a set valued function because it's a function. It's inverse, which is also a function, is then also a set-valued function.

Non-trivially, take any proper (i.e. non-injective) surjection from the set A to the set B. This would then be a "proper" set valued function, since it would then have multiple images corresponding to some (or all) of it's pre-images.

-Adam V. Nease.