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A correspondence f between X and Y is a triple (X,Y,Γ) where Γ is a subset of the Cartesian product X×Y.
The graph of a correspondence is not constrained in the number of edges into or out of each element: the image of a domain element can be empty, a singleton, or larger,
as can the pre-image of a range element.A function f:X→Y is a correspondence (X,Y,Γ) that assigns at most one range element to each element of its domain. Equivalently, the image of every element in the domain of f is a singleton.
A mapping is a function whose domain is its entire pre-domain.
A mapping f:X→Y is:
onto | or | surjective | if each y∈Y has | at least | one pre-image in f |
one-to-one | injective | at most | |||
one-to-one and onto | bijective | exactly |
The pre-domain and co-domain are not relevant for a relation,, since a relation is simply a set of tuples.
A relation r that is a subset of X×Y is:
functional | if each x∈X maps to | at most one | y∈Y |
injective | if each y∈Y is mapped to by | x∈X |