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Concept
Subset
Summary
In mathematics, 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. A k-subset is a subset with k elements.
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:
A is a subset of B, denoted by , or equivalently,
B is a superset of A, denoted by
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:
A is a proper (or strict) subset of B, denoted by , or equivalently,
B is a proper (or strict) superset of A, denoted by .
The empty set, written or is a subset of any set X and a proper subset of any set except itself, the inclusion relation is a partial order on the set (the power set of S—the set of all subsets of S) defined by . We may also partially order by reverse set inclusion by defining
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
suppose that a is a particular but arbitrarily chosen element of A
show that a is an element of B.
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.
The set of all subsets of is called its powerset, and is denoted by .
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