# Injections, Surjections, and Bijections

## Prerequisites

### Injections, Surjections, and Bijections

#### Introduction

• Functions can be classified by how they map elements between the domain and codomain.
• Three types of mappings are:
• Injective
• Surjective
• Bijective

#### Injective Mapping

• An injective mapping is when each element in the codomain is mapped to by at most one element from the domain.
• Another way of saying this is that no two elements in the domain map to the same element in the codomain.
• Example of an injective mapping:
• This type of mapping is also called 'one-to-one'.

#### Surjective Mapping

• A surjective mapping is when each element in the codomain is mapped to by at least one element from the domain.
• Another way of saying this is that the codomain and image (range) of the function are the same.
• Example of a surjective mapping:
• This type of mapping is also called 'onto'.

#### Bijective Mapping

• A bijective mapping is when the mapping is both injective and surjective.
• Another way of saying this is that each element in the codomain is mapped to by exactly one element in the domain.
• Example of a bijective mapping:
• This type of mapping is also called a 'one-to-one correspondence'.