Peirce’s 1870 “Logic Of Relatives” • Comment 11.7

We come now to the special cases of dyadic relations known as functions. It will serve a dual purpose in the present exposition to take the class of functions as a source of object examples for clarifying the more abstruse concepts of Relation Theory.

To begin, let us recall the definition of a local flag $L_{a @ j}$ of a $k$ -adic relation $L.$

$L_{a @ j} = \{ (x_1, \ldots, x_j, \ldots, x_k) \in L : x_j = a \}.$

In the case of a dyadic relation $L \subseteq X_1 \times X_2 = X \times Y,$ it is possible to simplify the notation for local flags in a couple of ways. First, it is often more convenient in the dyadic case to refer to $L_{u @ 1}$ and $L_{v @ 2}$ as $L_{u @ X}$ and $L_{v @ Y},$ respectively. Second, the notation may be streamlined even further by making the following definitions:

$\begin{array}{lllll} u \star L & = & L_{u @ X} & = & L_{u @ 1} \\[6pt] L \star v & = & L_{v @ Y} & = & L_{v @ 2} \end{array}$

In light of these conventions, the local flags of a dyadic relation $L \subseteq X \times Y$ may be comprehended under the following descriptions:

$\begin{array}{lll} u \star L & = & L_{u @ X} \\[6pt] & = & \{ (u, y) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that are incident with}~ u \in X. \\[9pt] L \star v & = & L_{v @ Y} \\[6pt] & = & \{ (x, v) \in L \} \\[6pt] & = & \text{the ordered pairs in}~ L ~\text{that are incident with}~ v \in Y. \end{array}$

The following definitions are also useful:

$\begin{array}{lll} u \cdot L & = & \mathrm{proj}_2 (u \star L) \\[6pt] & = & \{ y \in Y : (u, y) \in L \} \\[6pt] & = & \text{the elements of}~ Y ~\text{that are}~ L\text{-related to}~ u. \\[9pt] L \cdot v & = & \mathrm{proj}_1 (L \star v) \\[6pt] & = & \{ x \in X : (x, v) \in L \} \\[6pt] & = & \text{the elements of}~ X ~\text{that are}~ L\text{-related to}~ v. \end{array}$

A sufficient illustration is supplied by the earlier example $E.$

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The local flag $E_{3 @ X}$ of $E$ is displayed here:

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The local flag $E_{2 @ Y}$ of $E$ is displayed here:

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