[CG] Re: Triadic Relations

Cf: Triadic Relations • 2 https://inquiryintoinquiry.com/2021/11/08/triadic-relations-2/ Examples from Mathematics ========================= For the sake of topics to be taken up later, it is useful to examine a pair of triadic relations in tandem. We will construct two triadic relations, L₀ and L₁, each of which is a subset of the same cartesian product X × Y × Z. The structures of L₀ and L₁ can be described in the following way. Each space X, Y, Z is isomorphic to the boolean domain B = {0, 1} so L₀ and L₁ are subsets of the cartesian power B × B × B or the boolean cube B³. The operation of boolean addition, + : B × B → B, is equivalent to addition modulo 2, where 0 acts in the usual manner but 1 + 1 = 0. In its logical interpretation, the plus sign can be used to indicate either the boolean operation of exclusive disjunction or the boolean relation of logical inequality. The relation L₀ is defined by the following formula. • L₀ = { (x, y, z) ∈ B³ : x + y + z = 0 }. The relation L₀ is the following set of four triples in B³. • L₀ = { (0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0) }. The relation L₁ is defined by the following formula. • L₁ = { (x, y, z) ∈ B³ : x + y + z = 1 }. The relation L₁ is the following set of four triples in B³. • L₁ = { (0, 0, 1), (0, 1, 0), (1, 0, 0), (1, 1, 1) }. The triples in the relations L₀ and L₁ are conveniently arranged in the form of relational data tables, as shown below. Figure 0. Triadic Relation L₀ https://inquiryintoinquiry.files.wordpress.com/2020/05/triadic-relation-l0-… Figure 1. Triadic Relation L₁ https://inquiryintoinquiry.files.wordpress.com/2020/05/triadic-relation-l1-… References ========== Boolean Domain https://oeis.org/wiki/Boolean_domain Exclusive Disjunction https://oeis.org/wiki/Exclusive_disjunction Regards, Jon