Osajoukko

Osajoukko is the Finnish term for a subset in set theory. A is an osajoukko of B if every element of A is also an element of B. The standard notation to express this is A ⊆ B. If in addition A ≠ B, then A is a proper subset of B, sometimes written A ⊂ B depending on convention.

Key properties: For any set A, A ⊆ A (reflexivity). If A ⊆ B and B ⊆ C, then A ⊆ C (transitivity). If A ⊆ B and B ⊆ A, then A = B (antisymmetry). The empty set ∅ is a subset of every set, and every set is a subset of itself. The collection of all subsets of a given set S forms its power set P(S). If A ⊆ B, then the cardinality |A| ≤ |B|.

Examples: Let A = {1, 2} and B = {1, 2, 3, 4}. Then A ⊆ B. Let C = {1, 2} and D = {1, 2, 3}. Then C ⊊ D (proper subset).

Applications: Subsets underpin many basic constructions in mathematics, including intersections and unions, the definition of functions on domains, and concepts in probability and logic where containment and inclusion are central.

See also: subset, power set, membership, superset.