Triadic relation
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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 like 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.
[edit] 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) : xj 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:
| x | y | z |
|---|---|---|
| 0 | 0 | 0 |
| 0 | 1 | 1 |
| 1 | 0 | 1 |
| 1 | 1 | 0 |
