mtuples

mtuples refers to the collection of m-tuples, a fundamental concept in mathematics, computer science, and data modeling. An m-tuple is an ordered sequence of m elements drawn from one or more sets, commonly denoted as (x1, x2, ..., xm). When all elements come from the same set S, the set of all m-tuples from S is the Cartesian product S^m. If the components come from different sets S1, S2, ..., Sm, the set of all such m-tuples is S1 × S2 × ... × Sm.

Key properties include that order matters, elements may repeat, and the length of the sequence is fixed at m. For a finite set S with |S| = n, there are n^m possible m-tuples from S. If the component sets have sizes n1, n2, ..., nm, the total number of m-tuples is the product n1 × n2 × ... × nm.

In relation to vectors, an m-tuple can be viewed as a vector of length m with components in a given field or set, though in some contexts vectors carry additional structure such as a geometric or algebraic interpretation. In database theory, a relation of arity m is defined by its m-tuples, which correspond to rows with m attributes.

Common notational conventions include writing an m-tuple as (a1, a2, ..., am). The concept extends to higher-order constructs and underpins many combinatorial and probabilistic analyses, as well as modeling in programming and data systems. See also Cartesian product, n-tuple, k-tuple, and tuples in relational databases.