The set A is a subset of the set B if every element of A is an element of B. When this is so, A is contained in B, written A ⊆ B, and B contains A, written B ⊇ A. The following properties hold:
(i) For all sets A, Ø ⊆ A and A ⊆ A, where Ø denotes the empty set.
(ii) For all sets A and B, A = B if and only if A ⊆ B and B ⊆ A.
(iii) For all sets A, B and C, if A ⊆ B and B ⊆ C, then A ⊆ C.
See alsoalgebra of sets, power set, proper subset.
Computer
(of a set S) A set T whose members are all members of S; this is usually expressed as
A subset T is a proper subset of S if there is some element in S that is not in T; this is expressed as