## Peirce’s 1870 “Logic of Relatives” • Comment 11.9

### Peirce’s 1870 “Logic of Relatives” • Comment 11.9

Among the variety of regularities affecting dyadic relations we pay special attention to the $c$-regularity conditions where $c$ is equal to $1.$

Let $P \subseteq X \times Y$ be an arbitrary dyadic relation.  The following properties can be defined.

We previously examined dyadic relations exemplifying each of these regularity conditions.  Then we introduced a few bits of terminology and special-purpose notations for working with tubular relations.

We arrive by way of this winding stair at the special cases of dyadic relations $P \subseteq X \times Y$ variously described as $1$-regular, total and tubular, or total prefunctions on specified domains, $X$ or $Y$ or both, and which are more often celebrated as functions on those domains.

If $P$ is a pre-function $P : X \rightharpoonup Y$ that happens to be total at $X,$ then $P$ is known as a function from $X$ to $Y,$ typically indicated as $P : X \to Y.$

To say that a relation $P \subseteq X \times Y$ is total and tubular at $X$ is to say that $P$ is $1$-regular at $X.$  Thus, we may formalize the following definitions.

For example, let $X = Y = \{ 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$ and let $F \subseteq X \times Y$ be the dyadic relation depicted in the bigraph below.

$\text{Figure 39. Dyadic Relation}~ F$

We observe that $F$ is a function at $Y$ and we record this fact in either of the manners $F : X \leftarrow Y$ or $F : Y \to X.$

### Resources

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