Finitary Relation - Analogy With Functions

Analogy With Functions

A binary relation R on sets X and Y may be considered to associate, with each member of X, zero or more members of Y. (In the case of a relation T on more than two sets, X or Y or both can be cross products of any of the sets on which T is defined.) X is then referred to a the domain of R. Y is called the range or codomain of R. The subset of Y associated with a member x of X, is called the image of x, written as R(x). The subset of Y associated with a subset ξ of X is the union of the images of all the x in ξ and is called the image of ξ, written as R(ξ).

R is fully defined or total at X, if for every member x of X, there is at least one member y of Y where xRy. R is uniquely defined or tubular at X, if for every member x of X, there is at most one member y of Y where xRy. R is surjective or total at Y, if for every member y of Y, there is at least one member x of X where xRy. R is injective or tubular at Y, if for every member y of Y, there is at most one member x of X where xRy. If R is both fully defined and uniquely defined then R is well defined or 1-regular at X (for every member x of X, there is one and only one member y of Y where xRy). If R is both surjective and injective then R is bijective or 1-regular at Y. If R is both uniquely defined and injective then R is one-to-one.

A function is a well defined relation. A uniquely defined relation is a partial function. A surjective function is a surjection. An injective function is an injection. A bijective function is a bijection.

Relations generalize functions. Just as there is composition of functions, there is composition of relations.

Every binary relation R has a transpose relation R−1, which is related to the inverse function. For a relation R that is both fully defined and injective, the transpose relation R−1 is a true inverse in that R−1 faithfully restores any element x or subset ξ: R−1(R(ξ)) = ξ.