Subset and superset

In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is 'contained' inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. A is a subset of B may also be expressed as B includes A; or A is included in B.A is a proper subset of BC is a subset but not a proper subset of B In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is 'contained' inside B, that is, all elements of A are also elements of B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. A is a subset of B may also be expressed as B includes A; or A is included in B. The subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the subset relation is called inclusion. If A and B are sets and every element of A is also an element of B, then: If A is a subset of B, but A is not equal to B (i.e. there exists at least one element of B which is not an element of A), then For any set S, the inclusion relation ⊆ is a partial order on the set P ( S ) {displaystyle {mathcal {P}}(S)} of all subsets of S (the power set of S) defined by A ≤ B ⟺ A ⊆ B {displaystyle Aleq Biff Asubseteq B} . We may also partially order P ( S ) {displaystyle {mathcal {P}}(S)} by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A . {displaystyle Aleq Biff Bsubseteq A.} When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}.