Image (mathematics)

For the function that maps a Person to their Favorite Food, the image of Gabriela is Apple. The preimage of Apple is the set {Gabriela, Maryam}. The preimage of Fish is the empty set. The image of the subset {Richard, Maryam} is {Rice, Apple}. The preimage of {Rice, Apple} is {Gabriela, Richard, Maryam}.

In mathematics, the image of a function ⁠⁠ is the set of all ⁠⁠ such that ⁠⁠ belongs to the domain of ⁠⁠. The image by ⁠⁠ of an element ⁠⁠ of the domain of ⁠⁠ is ⁠⁠, that is, the output corresponding to the input ⁠⁠. The image by ⁠⁠ of a subset ⁠⁠ of the domain of ⁠⁠ is the set of all ⁠⁠ such that ⁠⁠ is in ⁠⁠, that is, the set of the images of the elements of ⁠⁠. Equivalently, it is the image of the restriction of ⁠⁠ to ⁠⁠.

Preimages or inverse images are defined similarly, by exchanging the roles of the domain and the codomain:

The preimage of an element ⁠⁠ of the codomain of ⁠⁠ is the set of all elements ⁠⁠ of the domain of ⁠⁠ such that ⁠⁠; it is empty if ⁠⁠ does not belong to the image of ⁠⁠. The preimage of a subset ⁠⁠ of the codomain of ⁠⁠ is the set of all elements ⁠⁠ of the domain of ⁠⁠ such that ⁠⁠. The preimage of the codomain of ⁠⁠ is, by definition of a function, the domain of ⁠⁠.

Images and inverse images may also be defined similarly for general binary relations. in this generalization, images and preimages play symmetric roles: the images and the preimages of a relation are respectively the preimages and the images of the opposite relation.

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