Function

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An example function between X  = {1, 2, 3} and Y  = {A, B, C, D}. This is neither injective nor surjective.

A function is a correspondence between two sets , where every element of corresponds to exactly one element of . Such a function can be denoted as , and the element that corresponds to is denoted by .

Simple examples of functions are constant functions satisfying for any , and the identity function which satisfies for all .

Glossary[edit | edit source]

The following are common terms used with or to describe functions:

  • If and are two functions, we may define their composition through the equation .

  • If different inputs give different outputs, the function is said to be injective.

  • If all values in are attained as outputs, the function is said to be surjective.

  • A function that is both injective and surjective is called bijective.

  • If a function is bijective, then there exists an inverse function such that .

  • A function where is an involution.

  • For a function and a set the image of a function on A , written , is the set of all for a  in A .

  • For a function and a set the preimage of a function on A , written , is the set of all a  such that is in A . Despite what the notation may imply, it is not necessary that f  have an inverse in order to take the preimage of a set.

  • A fixed point of a function, , is any x  such that f (x )=x .

External links[edit | edit source]