Function one-to-one and onto from set to an other

Question

I saw somewhere that if we have a one-to-one function from sets X to Y mean that we have a onto function from Y to X. I can't understand it !! Someone can explain ??

Answer 1

A function F: X → Y is into (aka injective) if every element of X is mapped to a distinct element of Y:

∀ x ∈ X, ∃ y ∈ Y | f(x) = y; x₁ ≠ x₂ ⇒ f(x₁) ≠ f(x₂)

It is onto (aka surjective) if every element of Y has some element of X that maps to it:

∀ y ∈ Y, ∃ x ∈ X | y = f(x)

And for F to be one-to-one (aka bijective), both of these things must be true. Therefore, by definition a one-to-one function is both into and onto.

But you say "an onto function from Y to X must exist." The "from Y to X" part might be what's tripping you up? F is onto, but it's from X to Y. The onto function from Y to X is F's inverse. Which must also be bijective, and therefore onto.

Some authors use "one-to-one" as a synonym for "injective" rather than "bijective". This disagreement is confusing, but we're stuck with it. However, under either definition, the inverse of F exists (every injective function has an inverse) and is surjective (F is defined for every element of X, therefore the inverse of F maps some element of Y to every element of X).

Answer 2

We can visualize this by drawing two circles, representing X and Y. The dots in the circle represent the elements in each set.

The arrows represent your function or "mapping".

So 1-1 means that every dot in the X circle maps to a unique dot in the Y circle.

Onto means that every dot has an arrow going to it. If you look at the picture, X is clearly not onto Y. There are two dots with no arrows coming in.

Now look at the "reverse" mapping by flipping the arrows on the lines.

Notice how in the reverse transform, every element of X has at least one element from Y going to it? That's the answer to your question. The 1-1 in the first picture (X to Y) means the second picture (Y to X) must be onto.

The wikipedia article on Surjective Functions explains this further.