## Injective, Surjective & Bijective

Table of Contents

## Definition of Function

A General Function

- Points each member of “A” to a member of “B”.

- A member of “A” only points one member of “B”. (
**one-to-many**is not allowed. Example: f(x) = 3 or 4 is not allowed)

- 2 or more members of “A” can point to the same “B” (
**many-to-one**is allowed. Example: f(4) = f(5) = 4 is allowed)

## Injective

Injective means

- No 2 or more members of “A” point to the same “B”.

**Many-to-one**is**NOT**allowed (Example: f(3) = f(4) = 2 is NOT allowed).

**One-to-many**is**NOT**allowed too (Because it’s a function)

- Member(s) of “B” without a matching “A” is allowed

- Injective is also called “
**One-to-One**“

A function

fis injectiveif and only ifwheneverf(x) = f(y), x = y.

## Surjective

Surjective means

- Every member of “B” has at least 1 matching “A” (can has more than 1).

- Member(s) of “B” without a matching “A” is
**NOT**allowed

f is surjective

if and only iff(A) = BA function

f(from setAtoB) is surjectiveif and only iffor everyyinB, there is at least one x inAsuch thatf(x) = y

## Bijective

Bijective means

- Both Injective and Surjective together.

- A perfect “
**one-to-one correspondence**” between the members of the sets. (Don’t get that confused with “One-to-One” used in injective).

A function

f(from setAtoB) is bijective if, for everyyinB, there isexactly onexinAsuch thatf(x) = y

## Inverse

Bijective means

Bijection function is also known as invertible function because it has inverse function property.

It is a function which assigns to

b, a unique elementasuch that f(a) =b. hence f^{-1}(b) =a.