combinesjoukko

Combinesjoukko is a term used in Finnish mathematical discourse to denote a set formed by applying a prescribed combination rule to tuples drawn from multiple source sets. The construction is abstract and can express many familiar ideas, including Cartesian products and various kinds of derived sets.

Formally, let I be a finite index set, S_i for i in I be sets, and R a rule mapping the Cartesian product ∏_{i∈I} S_i to a target domain U. The combinesjoukko C associated with (S_i, R) is defined as

C = { R(a_i) : (a_i) ∈ ∏_{i∈I} S_i }.

If R is the identity on the tuple, C is the Cartesian product S_1 × S_2 × ... × S_k. If R projects onto one component, C coincides with that component, and other choices of R yield various derived constructions.

Examples help illustrate the idea. Let A = {1, 2} and B = {x, y}. With R(a, b) = (a, b), the combinesjoukko is C = { (1, x), (1, y), (2, x), (2, y) }. If R concatenates symbols to form strings, the same A and B give C = { "1x", "1y", "2x", "2y" }. If R is addition, with A = {1, 2} and B = {3, 4}, one gets C = {4, 5, 5, 6}.

Properties include that C is finite if all S_i are finite; its size satisfies |C| ≤ ∏ |S_i|, with equality when R is injective on the product. If R maps distinct tuples to the same element, duplicates occur.

See also: Cartesian product, function application, Minkowski sum (in Euclidean spaces), and other set-construction methods. The term reflects the Finnish word joukko meaning “set.”