triples

Triples are ordered 3-tuples, commonly written as (a, b, c). Each component comes from a possibly different set, so a ∈ A, b ∈ B, c ∈ C, and the triple is an element of the Cartesian product A×B×C. Because order matters, (a, b, c) is not the same as (b, a, c). A triple can also be viewed as a point in three-dimensional space, such as a vector (x, y, z) in R^3.

In mathematics and computer science, triples appear in many models. In geometry and algebra, they specify coordinates, vectors, or parameters. In data semantics, RDF uses subject-predicate-object triples to express a statement; each triple encodes a basic fact. In combinatorics, triple systems are collections of 3-element subsets of a finite set, with Steiner triple systems being a notable example used in experimental design.

Operations on triples include projection to a component (taking the first, second, or third element) and combining triples via Cartesian products or function application. Triples differ from unordered 3-element sets, where the arrangement of elements is irrelevant; such sets are called 3-subsets or 3-element subsets. The terms "tuple" and "triple" are sometimes used interchangeably, but "triple" commonly emphasizes the trio as an ordered or relational unit in specific contexts, whereas "tuple" is a broader term in mathematics and computer science.

Related concepts include k-tuples for any fixed length k, and higher-dimensional analogues. The concept of a triple thus underlies representations in coordinates, databases, logic, and combinatorial design, making it a basic building block across disciplines.