### 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.

This site uses Akismet to reduce spam. Learn how your comment data is processed.