In mathematics, especially in set theory, 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.
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 of all subsets of S (the power set of S) defined by . We may also partially order by reverse set inclusion by defining .
When quantified, A ? B is represented as: ?x{x?A -> x?B}.^{[1]}
Formally:
Some authors use the symbols ? and ? to indicate subset and superset respectively; that is, with the same meaning and instead of the symbols, ? and ?.^{[2]} So for example, for these authors, it is true of every set A that .
Other authors prefer to use the symbols ? and ? to indicate proper subset and superset, respectively, instead of ? and ?.^{[3]} This usage makes ? and ? analogous to the inequality symbols x y 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.
Another example in an Euler diagram:
Inclusion is the canonical partial order in the sense that every partially ordered set (X, ) 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 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 (which is to say with each element of 2^{S}) the k-tuple from {0,1}^{k} of which the ith coordinate is 1 if and only if s_{i} is a member of T.
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