Cf: Relations & Their Relatives • 3 https://inquiryintoinquiry.com/2015/02/18/relations-their-relatives-3/
All, 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 notation is i|j. Table 1. Elementary Relatives for the “Divisor Of” Relation https://inquiryintoinquiry.files.wordpress.com/2015/02/elementary-relatives-for-the-e2809cdivisor-ofe2809d-relation.png 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.) Table 2. Logical Matrix for the “Divisor Of” Relation https://inquiryintoinquiry.files.wordpress.com/2015/02/logical-matrix-for-the-e2809cdivisor-ofe2809d-relation.png Just as matrices in linear algebra represent linear transformations, these logical arrays and matrices represent logical transformations. Resources ========= • Relation Theory ( https://oeis.org/wiki/Relation_theory ) • Triadic Relations ( https://oeis.org/wiki/Triadic_relation ) • Sign Relations ( https://oeis.org/wiki/Sign_relation ) • Peirce’s 1870 Logic Of Relatives ( https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview ) Regards, Jon
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