Properties of maps between sets

Consider two sets and .

Definition

A map from to is an assignment to each element of of an element of . The element that is assigned to is usually denoted .

For example, functions defined in high school are maps from the domain to the real numbers : they input a number from the domain and output a real number.

Definition

A map is called injective if

Intuition: Injective maps do not “glue” together elements, they “embed” into .

Definition

A map is called surjective if

Intuition: Surjective maps “fill” the entirety of , all elements of are “covered” by the map from .

Definition

A map is called bijective, or a one-to-one correspondence if it is both injective and surjective.