Cf: Triadic Relations • 1
At: http://inquiryintoinquiry.com/2020/05/21/triadic-relations-%e2%80%a2-1/
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.
Triadic Relation L₀
https://inquiryintoinquiry.files.wordpress.com/2020/05/triadic-relation-l0-bit-sum-0.png
Triadic Relation L₁
https://inquiryintoinquiry.files.wordpress.com/2020/05/triadic-relation-l1-bit-sum-1.png
Resources
=========
• Logic Syllabus ( https://inquiryintoinquiry.com/logic-syllabus/ )
• Sign Relations ( https://oeis.org/wiki/Sign_relation )
• Triadic Relations ( https://oeis.org/wiki/Triadic_relation )
• Relation Theory ( https://oeis.org/wiki/Relation_theory )
Regards,
Jon
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