What is mathimatical domain range?

Answer 1

Take a map S from set A to set B, denote S: A ---> B

We call A to be our domain, B our codomain.

We call, with an small abuse of notation, S(A) our range, that is the set of all maps of elements of A. Or, we call set C the range of S if C = {c | c = S(a) for all a from A}

Remark, C is a subset of B.

Just for further knowledge, if for all a in A, S(a) is different, or S(a) != S(b) => a != b for all a, b from A (S(a) and S(b) from B), then we say S is one-to-one. It can happen when the "size" of A is smaller or equal than that of B.

if the range of S is the same as the codomain. Or for all elements c from B, c = S(a) for some a from A, then we call S to be onto. It can happen when "size" of A is larger or equal to that of B.

Further, if S is one-to-one AND onto, it is invertible. I will leave the proof as an exercise.

Just two more note:

1. S is linear if it's a map between vector space A, B over field F which also satisfies S(a + b) = S(a) (+) S(b) and S(kb) = k.S(b) where + and x are addition and scalar multiplication from A while (+) and . are for B.

2. S is not necessarily a function.