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

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 $L.$

For a dyadic relation $L \subseteq X \times Y$ the notation for local flags can be simplified in two ways.  First, the local flags $L_{u @ 1}$ and $L_{v @ 2}$ are often more conveniently notated as $L_{u @ X}$ and $L_{v @ Y},$ respectively.  Second, the notation may be streamlined even further by making the following definitions.

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

The following definitions are also useful.

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

$\text{Figure 35. Dyadic Relation}~ E$

Figure 36 shows the local flag $E_{3 @ X}$ of $E.$

$\text{Figure 36. Local Flag}~ E_{3 @ X}$

Figure 37 shows the local flag $E_{2 @ Y}$ of $E.$

$\text{Figure 37. Local Flag}~ E_{2 @ Y}$

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