### 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.

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