Proper subset

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 ${\displaystyle {\mathcal {P}}(S)}$ of all subsets of S (the power set of S) defined by ${\displaystyle A\leq B\iff A\subseteq B}$. We may also partially order ${\displaystyle {\mathcal {P}}(S)}$ by reverse set inclusion by defining ${\displaystyle A\leq B\iff B\subseteq A}$.

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