Sets and applications

Applications

Let be sets. A function assigns to each a unique element . Functions are also called maps. mappings or application.

Definition

Let be a function. Then is called the domain of and is called the codomain of . We write to indicate that is the function that maps to .

Example

Let and . Then we can define a function by setting and . is the image of inder , is the perimage of under .

Injective function

The function is injective, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain.

Definition

A function is injective if we have: . An injection is also know as a one to one function.

Example

  • The function defined by is not injective since

  • The function defined by is injective.

Surjective function

The function is surjective, if each element of the codomain is mapped to by at least one element of the domain. That is, the image and the codomain of the function are equal.

Definition

A function is surjective if we have .

Example

The function is surjective.

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