Sets and functionsFunction Properties
According to the A function from a set to another set is an assignment of some element of to each element in . The codomain of a function from one set to another set is the set . The codomain must be specified as part of the function's definition, and there is no guarantee that every element of the codomain has any preimages in the domain. Codomain is similar to range, but the range of a function includes only those elements of the codomain which have preimages in the domain.
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.
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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.