Subset

In mathematics, a set "A" is a subset of a set "B" if all elements of "A" are also elements of "B"; "B" is then a superset of "A". It is possible for "A" and "B" to 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 (or contains) "A" or "A" is included (or contained) 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.
Definitions.
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_7 (the power set of "S"—the set of all subsets of "S") defined by formula_8. We may also partially order formula_7 by reverse set inclusion by defining formula_10
When quantified, is represented as .
We can prove the statement by applying a proof technique known as the element argument:Let sets "A" and "B" be given. To prove that ,
The validity of this technique can be seen as a consequence of Universal generalization: the technique shows for an arbitrarily chosen element "c". Universal generalisation then implies , which is equivalent to , as stated above.
⊂ and ⊃ symbols.
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 <. For example, if , then "x" may or may not equal "y", but if , then "x" definitely does not equal "y", and "is" less than "y". Similarly, using the convention that ⊂ is proper subset, if , then "A" may or may not equal "B", but if , then "A" definitely does not equal "B".
Examples of subsets.
Another example in an Euler diagram:
Other properties of inclusion.
Inclusion is the canonical partial order, in the sense that every partially ordered set ("X", formula_14) is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example: if each ordinal "n" is identified with the set ["n"] of all ordinals less than or equal to "n", then "a" ≤ "b" if and only if ["a"] ⊆ ["b"].
For the power set formula_7 of a set "S", the inclusion partial order is—up to an order isomorphism—the Cartesian product of "k" = |"S"| (the cardinality of "S") copies of the partial order on {0,1} for which 0 < 1. This can be illustrated by enumerating "S" = {"s"1, "s"2, ..., "s""k"}, and associating with each subset "T" ⊆ "S" (i.e., 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".