Triadic Relations • 2

Triadic Relations • 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, $L_0$ and $L_1,$ each of which is a subset of the same cartesian product $X \times Y \times Z.$ The structures of $L_0$ and $L_1$ can be described in the following way.

Each space $X, Y, Z$ is isomorphic to the boolean domain $\mathbb{B} = \{ 0, 1 \}$ so $L_0$ and $L_1$ are subsets of the cartesian power $\mathbb{B} \times \mathbb{B} \times \mathbb{B}$ or the boolean cube $\mathbb{B}^3.$

The operation of boolean addition, $+ : \mathbb{B} \times \mathbb{B} \to \mathbb{B},$ 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 $L_0$ is defined by the following formula.

$L_0 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 0 \}.$

The relation $L_0$ is the following set of four triples in $\mathbb{B}^3.$

$L_0 ~=~ \{ ~ (0, 0, 0), ~ (0, 1, 1), ~ (1, 0, 1), ~ (1, 1, 0) ~ \}.$

The relation $L_1$ is defined by the following formula.

$L_1 ~=~ \{ (x, y, z) \in \mathbb{B}^3 : x + y + z = 1 \}.$

The relation $L_1$ is the following set of four triples in $\mathbb{B}^3.$

$L_1 ~=~ \{ ~ (0, 0, 1), ~ (0, 1, 0), ~ (1, 0, 0), ~ (1, 1, 1) ~ \}.$

The triples in the relations $L_0$ and $L_1$ are conveniently arranged in the form of relational data tables, as shown below.

Document History

Portions of the above article were adapted from the following sources under the GNU Free Documentation License, under other applicable licenses, or by permission of the copyright holders.

cc: Category Theory • Cybernetics (1) (2) • Ontolog (1) (2)
cc: Structural Modeling (1) (2) • Systems Science (1) (2)
cc: FB | Relation Theory • Laws of Form • Peirce List (1) (2)

5 Responses to Triadic Relations • 2

Pingback: Survey of Relation Theory • 5 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 4 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 3 | Inquiry Into Inquiry
Pingback: Triadic Relations • Discussion 3 | Inquiry Into Inquiry
Pingback: Survey of Relation Theory • 6 | Inquiry Into Inquiry

	Survey of Animated L… on Logic Syllabus • Discussion…
	Logic Syllabus • Dis… on Logic Syllabus
	Survey of Animated L… on Logic Syllabus • Discussion…
	Survey of Animated L… on Logic Syllabus
	Logic Syllabus • Dis… on Genus, Species, Pie Charts, Ra…
	Logic Syllabus • Dis… on Logic Syllabus
	Survey of Inquiry Dr… on Inquiry Into Inquiry • On Init…
	Inquiry Into Inquiry… on Survey of Abduction, Deduction…
	Inquiry Into Inquiry… on Survey of Pragmatic Semiotic I…
	Inquiry Into Inquiry… on Survey of Inquiry Driven Syste…

M	T	W	T	F	S	S
1	2	3	4	5	6	7
8	9	10	11	12	13	14
15	16	17	18	19	20	21
22	23	24	25	26	27	28
29	30