Prove: A mapping f:S->T is bijective if and only if it has an inverse?

[mruncleramos]

Can a mapping from f:S->T associate an element of s into several elements of T? Also, how do you prove: A mapping f:S->T is bijective if and only if it has an inverse?

 

[HallsofIvy]

No. "mapping" is another word for "function". Part of the definition of function is that f(x) be a single value.

f:S-> T is bijective iplies f has an inverse:

Let y be any member of T. Since f is bijective it is both one-to-one (injective) and onto (surjective). Since f is surjective, there exist at least one member of S, x, such that f(x)= y. Since f is injective, there is only one such. Define g: T->S by g(y)= x.
Now show that that is a function and is the inverse of f.

f has an inverse implies f is bijective.

i) for any y in T, let x= f^-1(y). Then, by definition of inverse, f(x)= y so f is surjective.

ii) suppose f(x₁)= f(x₂). Applying f^-1 to both sides of the equation f^-1(f(x₁)= f^-1(x₂)
x₁= x₂
so f is injective.

Since f is both injective and surjective, it is bijective.

 

[kioria]

Can a mapping from f:S->T associate an element of s into several elements of T? Also, how do you prove: A mapping f:S->T is bijective if and only if it has an inverse?

To take things into an account, I remember in discrete maths - a bit like computational mathematics or rather, it is computational mathematics - we define set |R| = |{1,1,1,1}| as 4. As there are four items in the set. Now, in the other extreme, let S = {1}, then |S| = |{1}| = 1. So in effect the sets R and S have different properties.

One other thing is that, once again, in computer science, R = {1,1,1,1} and S = {1} are two totally different sets. Let's map (+2) to both sides and find the total sum of all the elements in the sets, i.e., {sum (map (+2) R)} and {sum (map (+2) S)}.

sum (map (+2) R) = sum {3,3,3,3} = 12
sum (map (+2) S) = sum {1} = 1 /= 12 = sum (map (+2) R)
and as function applied two both sets are equal, R and S are not equal.

However, answering your question, even in computer science the values of each element after mapping only takes 1 value from the image. i.e., notice that (map (+2) R) = {3,3,3,3}. So 1 -> 3 by mapping. And each element valued 1, goes to exactly one value in the image of f, that is 3.

But as I said, this is in computer science and most of the computer applications - although duplicates are considered nasty in some cases.

 

[matt grime]

The set {1,1,1,1} contains only one element. you aren't allowed to repeat elements in a set.

 

[kioria]

I meant cardinalities... yes my wordings were pretty poor.