Bijective Functions

A bijection is the "perfect pairing." It implies the existence of an inverse function, which is a powerful tool in solving equations.

Definition

Definition (Bijection)

A function

f

is bijective if it is both injective and surjective. There is a one-to-one correspondence between the domain and codomain.

f : A \to B

is bijective, there exists a unique inverse function

f^{- 1} : B \to A

such that:

The graph of

y = f^{- 1} (x)

is the reflection of

y = f (x)

across the line

y = x

In combinatorics, establishing a bijection between two sets proves they have the same size.

Example: The number of subsets of a set of size $n$ is equal to the number of binary strings of length $n$ .

Exercise (Problem 1)

Prove that

f (x) = 3 x - 5

is a bijection from

R \to R

and find its inverse.

Exercise (Problem 2)

Let

f (x) = \frac{x}{1 + ∣ x ∣}

. Prove that

f

is a bijection from

R

(- 1, 1)

Exercise (Problem 3)

f

and

g

are bijective, prove that the composition

f \circ g

is also bijective.