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.
Read more about this topic: Finitary Relation
Famous quotes containing the words analogy and/or functions:
“The analogy between the mind and a computer fails for many reasons. The brain is constructed by principles that assure diversity and degeneracy. Unlike a computer, it has no replicative memory. It is historical and value driven. It forms categories by internal criteria and by constraints acting at many scales, not by means of a syntactically constructed program. The world with which the brain interacts is not unequivocally made up of classical categories.”
—Gerald M. Edelman (b. 1928)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)