EnglishSets 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.
