## Relation Theory • 2

### Relation Theory • Preliminaries

Two definitions of the relation concept are common in the literature.  Although it is usually clear in context which definition is being used at a given time, it tends to become less clear as contexts collide, or as discussion moves from one context to another.

The same sort of ambiguity arose in the development of the function concept and it may save a measure of effort to follow the pattern of resolution that worked itself out there.

When we speak of a function $f : X \to Y$ we are thinking of a mathematical object whose articulation requires three pieces of data, specifying the set $X,$ the set $Y,$ and a particular subset of their cartesian product ${X \times Y}.$  So far so good.

Let us write $f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)$ to express what has been said so far.

When it comes to parsing the notation ${}^{\backprime\backprime} f : X \to Y {}^{\prime\prime},$ everyone takes the part ${}^{\backprime\backprime} X \to Y {}^{\prime\prime}$ as indicating the type of the function, in effect defining $\mathrm{type}(f)$ as the pair $(\mathrm{obj_1}f, \mathrm{obj_2}f),$ but ${}^{\backprime\backprime} f {}^{\prime\prime}$ is used equivocally to denote both the triple $(\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)$ and the subset $\mathrm{obj_{12}}f$ forming one part of it.

One way to resolve the ambiguity is to formalize a distinction between the function $f = (\mathrm{obj_1}f, \mathrm{obj_2}f, \mathrm{obj_{12}}f)$ and its graph, defining $\mathrm{graph}(f) = \mathrm{obj_{12}}f.$

Another tactic treats the whole notation ${}^{\backprime\backprime} f : X \to Y {}^{\prime\prime}$ as a name for the triple, letting ${}^{\backprime\backprime} f {}^{\prime\prime}$ denote $\mathrm{graph}(f).$

In categorical and computational contexts, at least initially, the type is regarded as an essential attribute or integral part of the function itself.  In other contexts we may wish to use a more abstract concept of function, treating a function as a mathematical object capable of being viewed under many different types.

Following the pattern of the functional case, let the notation ${}^{\backprime\backprime} L \subseteq X \times Y {}^{\prime\prime}$ bring to mind a mathematical object specified by three pieces of data, the set $X,$ the set $Y,$ and a particular subset of their cartesian product ${X \times Y}.$  As before we have two choices, either let $L$ be the triple $(X, Y, \mathrm{graph}(L))$ or let ${}^{\backprime\backprime} L {}^{\prime\prime}$ denote $\mathrm{graph}(L)$ and choose another name for the triple.

### Resources

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