Subset  In mathematics, a set "A" is a subset of a set "B", or equivalently "B" is a superset of "A", if "A" is contained in "B". That is, all elements of "A" are also elements of "B". "A" and "B" may be equal; if they are unequal, then "A" is a proper subset of "B". 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. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. 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 formula_5 of all subsets of "S" (the power set of "S") defined by formula_6. We may also partially order formula_5 by reverse set inclusion by defining formula_8  When quantified, is represented as . Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively; that is, with the same meaning and instead of the symbols, ⊆ and ⊇. For example, for these authors, it is true of every set "A" that . Other authors prefer to use the symbols ⊂ and ⊃ to indicate "proper" (also called strict) subset and "proper" superset respectively; that is, with the same meaning and instead of the symbols, ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and 1, "s"2, ..., "s""k"} and associating with each subset "T" ⊆ "S" (which is to say with each element of 2"S") the "k"-tuple from {0,1}"k" of which the "i"th coordinate is 1 if and only if "s""i" is a member of "T".