# Sets and functionsFunction Properties

According to the

**Definition**

A function is **injective** if no two elements in the domain map to the same element in the codomain; in other words if implies .

A function is **surjective** if the range of is equal to the codomain of ; in other words, if implies that there exists with .

A function is **bijective** if it is both injective and surjective. This means that for every , there is exactly one such that . If is bijective, then the **inverse** of is the function from to that maps to the element that satisfies .

**Exercise**

Identify each of the following functions as injective or not injective, surjective or not surjective, and bijective or not bijective.

- ,
and - ,
and - ,
and - ,
and

**Exercise**

For each of the four combinations of injectivity and surjectivity, come up with a real-world function which has that property.

For example, the function from the set of ticket numbers for a commercial airplane flight to the set of passengers on the plane (the one which associates each ticket number with the passenger named on that ticket) is bijective.