Triadic relation

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 nowrap|"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 nowrap|"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 = {("x"₁, "x"₂, "x"₃) : "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:

ee also

* Relation
* Logical matrix