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.

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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.

3 Responses to Relations & Their Relatives : 3

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

  2. Pingback: Survey of Relation Theory • 2 | Inquiry Into Inquiry

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

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