Triadic relation.html

In logic and mathematics, a triadic relation or a ternary relation is an important special case of a polyadic or finitary relation, one in which the number of places in the relation is three. One also sees the adjectives 3-adic, 3-ary, 3-dim, or 3-place being used to describe these relations.

Just as a binary relation is a set of pairs, forming a subset of some Cartesian product A× B of a pair of sets A and B, so a ternary relation is a set of triplets, forming a subset of the Cartesian product A × B × C of three sets A, B and C.

Example

The boolean domain is the set B = {0, 1}. The plus sign "+", used in the context of the boolean domain B, denotes addition mod 2. Interpreted for logic, this amounts to the same thing as the boolean operation of exclusive-or or not-equal-to.

The third Cartesian power of B is B³ = B × B × B = {(x₁, x₂, x₃) : x_j in B for j = 1, 2, 3}.

We define a subset L of B³.

The relation L is defined as follows:

L = {(x, y, z) in B³ : x + y + z = 0}.

The relation L is the set of four triples enumerated here:

L = {(0, 0, 0), (0, 1, 1), (1, 0, 1), (1, 1, 0)}.

The triples that make up the relation L are conveniently arranged in the form of a relational data table, as follows:

$L = \{(x, y, z) \in \mathbb{B}^3 : x + y + z = 0\}$
x	y	z
0	0	0
0	1	1
1	0	1
1	1	0

Triadic relation.html

Example

See also