- #1

- 27

- 0

## Homework Statement

Suppose f is bijection. Prove that f⁻¹. is bijection.

## Homework Equations

A bijection of a function occurs when f is one to one and onto.

I think the proof would involve showing f⁻¹. is bijective, by showing f⁻¹ is onto, and one to one, since f is bijective it is invertible.

## The Attempt at a Solution

To start:

Since f is invertible/bijective

f⁻¹ is one-to-one:

f:A→B

f⁻¹:B→A

f(a)=b then f⁻¹(b)=a

if f(a)=b f(a)=b' then b=b'

So f⁻¹ is one-to-one

f⁻¹ is onto:

*for f to be onto:

f:A→B

∀a∈A,∃b∈B f(a)=b

*

f⁻¹:B→A

∀b∈B,∃a∈A f(b)=a

So f⁻¹ is onto.

Therefore f⁻¹ is bijective.

Would this be correct?

Thank You Very Much.