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.


LOR 1870 Figure 30
(35)

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


LOR 1870 Figure 36
(36)

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


LOR 1870 Figure 37
(37)
This entry was posted in Graph Theory, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Relation Theory, Semiotics and tagged , , , , , , , . Bookmark the permalink.

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

  1. Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry

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