Relations & Their Relatives • 3

Posted on August 2, 2024 by Jon Awbrey

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 corresponding 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 mathematical notation is {i|j}.

Elementary Relatives for the “Divisor Of” Relation

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 for ease of reading.)

Logical Matrix for the “Divisor Of” Relation

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

Resources

cc: FB | Relation Theory • Laws of Form • Mathstodon • Academia.edu
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science

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