subsetrelation

The subset relation, denoted ⊆, is a relation between sets that states A ⊆ B if and only if every element of A is also an element of B. If A ⊆ B but A ≠ B, A is called a proper subset of B and is sometimes written A ⊊ B.

In set theory, the subset relation is a partial order on the class of all sets (and, in particular, on the power set P(U) of any universal set U). It is reflexive (A ⊆ A for all A), antisymmetric (if A ⊆ B and B ⊆ A then A = B), and transitive (if A ⊆ B and B ⊆ C then A ⊆ C).

The structure (P(U), ⊆), where P(U) is the set of all subsets of U, forms a complete lattice. The meet operation is intersection (A ∩ B), and the join operation is union (A ∪ B). The bottom element is the empty set ∅ and the top element is U. This lattice also has the structure of a Boolean algebra with complement relative to U.

Examples: If A = {1, 2} and B = {1, 2, 3}, then A ⊆ B. If A = {1} and B = {2}, neither A ⊆ B nor B ⊆ A holds. If A = B, then both A ⊆ B and B ⊆ A are true.

Applications of the subset relation include expressing inclusion hierarchies, defining monotone functions, reasoning about intersections and unions, and underpinning constructions such as the power set in various areas of mathematics and computer science.