Function as Rule

Types: Onto, One-To-One, Invertible
Examples: Not Applicable
Constructions: Image of Function, Inverse Image
Generalizations: Not Applicable

Properties: Composition
Sufficiencies: Not Applicable
Equivalences: Not Applicable
Justifications: Not Applicable

Function as Rule

A function consists of three things: two sets, called the Domain and the Codomain, and a rule that associates to any element in the domain exactly one element in the codomain.

Remark. Typically, we say “let be a function” to indicate that the domain is , the codomain is , and the rule is .

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Remark. It may be possible to evaluate the function on every element of the domain, and every output (value of the function) must be in the codomain. But it is not necessary that every element of the codomain be a value of the function. We use the word “image” to denote the set of elements in the codomain that are actually reached.

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Remark. The words function, mapping, and map are synonyms, generally used in different contexts. A function normally returns a number. Mapping was first used in topology and geometry and has spread to all parts of mathematics. In higher dimensions, we tend to use the word mapping rather than function.