tulojoukon

Tulojoukon is a term used in Finnish mathematics to denote the product set of two sets. If A and B are subsets of a ring R (commonly the real numbers), the tulojoukon AB is defined as AB = {ab : a ∈ A, b ∈ B}. It consists of all products formed by choosing one element from each set.

The concept is widely used in areas such as real analysis, algebra, and additive combinatorics. For finite sets A and B, AB is finite and its size satisfies |AB| ≤ |A|·|B|. If either A or B is empty, AB is empty. If 0 ∈ A or 0 ∈ B, then 0 ∈ AB. In general AB is a subset of R and need not have any particular simple shape.

Examples help illustrate: if A = {1, 2} and B = {3, 4}, then AB = {3, 4, 6, 8}. If A = {a} and B = {b} are singleton sets, AB = {ab}. For intervals, the shape of AB depends on the signs and ranges of A and B. If A and B lie in [0, ∞) (or in (−∞, 0]), the product AB is an interval given by the extreme products of endpoints. When A and B have elements of mixed signs, AB can be a union of multiple intervals.

Properties related to AB include basic algebraic rules such as distributivity over union: A(C ∪ D) = AC ∪ AD and (A ∪ B)C = AC ∪ BC, provided the products are formed in a compatible algebraic setting. The concept extends to more general rings and fields and is related to the study of multiplicative structures and semigroups.

See also: product set, Minkowski product, multiplicative semigroup.