In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of collinearity among three points in Euclidean space. In an abstract set, a ternary equivalence relation determines a collection of equivalence classes or pencils that form a linear space in the sense of incidence geometry. In the same way, a binary equivalence relation on a set determines a partition.
Definition
A ternary equivalence relation on a set X is a relation E ⊂ X3, written [a, b, c], that satisfies the following axioms:
- Symmetry: If [a, b, c] then [b, c, a] and [c, b, a]. (Therefore also [a, c, b], [b, a, c], and [c, a, b].)
- Reflexivity: [a, b, b]. Equivalently, if a, b, and c are not all distinct, then [a, b, c].
- Transitivity: If a ≠ b and [a, b, c] and [a, b, d] then [b, c, d]. (Therefore also [a, c, d].)
References
Ternary equivalence relation Wikipedia(Text) CC BY-SA