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# Sets and functionsFunction Properties

Czas czytania: ~10 min

According to the definition, a function may map several elements of the domain to the same element of the codomain, and it may also miss elements in the codomain (in other words, fail to map any domain element to them). However, these behaviors can be undesirable in some situations, and we'll want terminology to refer to whether a given function exhibits them.

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.

Bruno