Question

prove that only one-one onto
function. possesses an inverse function​

Answer 1

Answer:

Let f : A → B be bijective function. Then f posses an inverse.

Proof.

Let

f : A → B

be bijective.

We will define a inverse function

g : B → A

Let b ∈ B.

Since f is surjective, there exists a ∈ A such that

g(a) = b.

Let

g(b) = a.

Since f is injective, therefor a is unique, so g is well-defined.

Now To check that g is the inverse of f.

First we will show that

g ◦ f = A.

Let

a ∈ A. Let b = f(a). Then, by definition

g(b) = a.

Then

g ◦ f(a) = g(f(a)) = g (b) = a.

Which show that

g ◦ f = A.

Second we will show that

f ◦ g = B.

Let b ∈ B. Let

a = g(b). Then, by definition,

f(a) = b.

Then

f ◦ g(b) = f(g(b)) = f(a) = b.

Which show that

f ◦ g = B.

Hence Bijective mapping have a inverse