Codomain

https://en.wikipedia.org/wiki/Codomain https://proofwiki.org/wiki/Definition:Codomain_(Set_Theory)/Relation

The codomain of a function , or the target set, is the set into which all of the output of the function is constrained to fall.

It is the set Y in the notation f: X → Y.

The term range is sometimes ambiguously used to refer to either the codomain or image of a function.

A codomain is part of a function f if f is defined as a triple (X, Y, G) where X is called the domain of f, Y its codomain, and G its graph.

The set of all elements of the form f(x), where x ranges over the elements of the domain X, is called the image of f. The image of a function is a subset of its codomain so it might not coincide with it. Namely, a function that is not surjective has elements y in its codomain for which the equation f(x) = y does not have a solution.

A codomain is not part of a function f if f is defined as just a graph. For example in set theory it is desirable to permit the domain of a function to be a proper class X, in which case there is formally no such thing as a triple (X, Y, G). With such a definition functions do not have a codomain, although some authors still use it informally after introducing a function in the form f: X → Y.

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A codomain of a function is the target set in operations or relations on two sets.

Given two sets A and B, where ∀a ∈ A and ∀b ∈ B, set B is the codomain:

• in set operations (union, intersection, etc.), e.g. A U B = {a ∨ b}

• in the dot product, A x B = {(a,b)}

• in relations on sets, aRb i.e. {(a,b)} = R ⊆ A x B

• in functions, f :: A -> B and f = {(a,b)}

The codomain set is sometimes denoted by $B_{cod}$.