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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*. *A* and *B* may coincide. The relationship of one set being a subset of another is called **inclusion** or sometimes **containment**.

If *A* and *B* are sets and every element of *A* is also an element of *B*, then:

*A*is a subset of (or is included in)*B*, denoted by

*A**\subseteq**B*

or equivalently

*B*is a**superset**of (or includes)*A*, denoted by

*B**\supseteq**A.*

If *A* is a subset of *B*, but *A* is not equal to *B* (i.e. there exists at least one element of B not contained in *A*), then

*A*is also a**proper**(or**strict**) subset of*B*; this is written as

*A\subsetneq**B.*

or equivalently

*B*is a proper superset of*A*; this is written as

*B\supsetneq**A.*

For any set *S*, the inclusion relation ⊆ is a partial order on the set

l{P}(S)

Some authors use the symbols ⊂ and ⊃ to indicate "subset" and "superset" respectively, instead of the symbols ⊆ and ⊇, but with the same meaning. So for example, for these authors, it is true of every set *A* that *A* ⊂ *A*.

Other authors prefer to use the symbols ⊂ and ⊃ to indicate *proper* subset and superset, respectively, in place of ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if *x* ≤ *y* then *x* may be equal to *y*, or maybe not, but if *x* < *y*, then *x* definitely does not equal *y*, but is strictly less than *y*. Similarly, using the "⊂ means proper subset" convention, if *A* ⊆ *B*, then *A* may or may not be equal to *B*, but if *A* ⊂ *B*, then *A* is definitely not equal to *B*.

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