## Relations & Their Relatives : 3

Here are two ways of looking at the divisibility relation, a dyadic relation of fundamental importance in number theory.

Table 1 shows the first few ordered pairs of the relation on positive integers that corresponds to the relative term, “divisor of”.  Thus, the ordered pair ${i\!:\!j}$ appears in the relation if and only if ${i}$ divides ${j},$ for which the usual notation is ${i|j}.$

$\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 1. Elementary Relatives for the Divisor Of" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1\!\!:\!\!1&1\!:\!2&1\!:\!3&1\!:\!4&1\!:\!5&1\!:\!6&1\!:\!7&1\!:\!8&1\!:\!9&1\!:\!10&\dots \\ 2&&2\!:\!2&&2\!:\!4&&2\!:\!6&&2\!:\!8&&2\!:\!10&\dots \\ 3&&&3\!:\!3&&&3\!:\!6&&&3\!:\!9&&\dots \\ 4&&&&4\!:\!4&&&&4\!:\!8&&&\dots \\ 5&&&&&5\!:\!5&&&&&5\!:\!10&\dots \\ 6&&&&&&6\!:\!6&&&&&\dots \\ 7&&&&&&&7\!:\!7&&&&\dots \\ 8&&&&&&&&8\!:\!8&&&\dots \\ 9&&&&&&&&&9\!:\!9&&\dots \\ 10&&&&&&&&&&10\!:\!10&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}$

Table 2 shows the same information in the form of a logical matrix.  This has a coefficient of ${1}$ in row ${i}$ and column ${j}$ when ${i|j},$ otherwise it has a coefficient of ${0}.$  (The zero entries have been omitted here for ease of reading.)

$\begin{array}{|c||*{11}{c}|} \multicolumn{12}{c}{\text{Table 2. Logical Matrix for the Divisor Of" Relation}} \\[4pt] \hline i|j &1&2&3&4&5&6&7&8&9&10&\ldots \\ \hline\hline 1&1&1&1&1&1&1&1&1&1&1&\dots \\ 2& &1& &1& &1& &1& &1&\dots \\ 3& & &1& & &1& & &1& &\dots \\ 4& & & &1& & & &1& & &\dots \\ 5& & & & &1& & & & &1&\dots \\ 6& & & & & &1& & & & &\dots \\ 7& & & & & & &1& & & &\dots \\ 8& & & & & & & &1& & &\dots \\ 9& & & & & & & & &1& &\dots \\ 10&& & & & & & & & &1&\dots \\ \ldots&\ldots&\ldots&\ldots&\ldots&\ldots& \ldots&\ldots&\ldots&\ldots&\ldots&\ldots \\ \hline \end{array}$

Just as matrices in linear algebra represent linear transformations, these logical arrays and matrices represent logical transformations.

### Resources

Advertisements
This entry was posted in C.S. Peirce, Denotation, Logic, Logic of Relatives, Mathematics, Number Theory, Peirce, Relation Theory, Semiotics, Sign Relations, Triadic Relations and tagged , , , , , , , , , , . Bookmark the permalink.