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Projection (mathematics) – Mapping equal to its square under mapping composition; Projection (measure theory) Projection (linear algebra) – Idempotent linear transformation from a vector space to itself; Projection (relational algebra) – Operation that restricts a relation to a specified set of attributes
The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map is a homomorphism of vector spaces, while the term linear function may have this meaning or it may mean a linear polynomial. [3] [4] In category theory, a map may refer to a morphism. [2]
Generally, a mapping where the domain and codomain are the same set (or mathematical structure) is a projection if the mapping is idempotent, which means that a projection is equal to its composition with itself. A projection may also refer to a mapping which has a right inverse. Both notions are strongly related, as follows. Let p be an ...
In mathematics, if is a subset of , then the inclusion map is the function that sends each element of to , treated as an element of ::, =. An inclusion map may also be referred to as an inclusion function , an insertion , [ 1 ] or a canonical injection .
Moreover, f is the composition of the canonical projection from f to the quotient set, and the bijection between the quotient set and the codomain of . The composition of two surjections is again a surjection, but if g ∘ f {\displaystyle g\circ f} is surjective, then it can only be concluded that g {\displaystyle g} is surjective (see figure).
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories. Categories (such as subcategories of Top) without adjoined products may still have an ...
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}.
The empty set serves as the initial object in Set with empty functions as morphisms. Every singleton is a terminal object, with the functions mapping all elements of the source sets to the single target element as morphisms. There are thus no zero objects in Set. The category Set is complete and co-complete.