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
CONTENTS * 1 Definitions * 2 Property * 3 ⊂ and ⊃ symbols * 4 Examples * 5 Other properties of inclusion * 6 See also * 7 References * 8 External links DEFINITIONS 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 B {displaystyle Asubseteq B} , or equivalently * B is a SUPERSET of (or includes) A, denoted by B A . {displaystyle Bsupseteq 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 which is not an element of A), then * A is also a PROPER (or STRICT) SUBSET of B; this is written as A B . {displaystyle Asubsetneq B.} or equivalently * B is a PROPER SUPERSET of A; this is written as B A . {displaystyle Bsupsetneq A.} For any set S, the inclusion relation ⊆ is a partial order on the set P ( S ) {displaystyle {mathcal {P}}(S)} of all subsets of S (the power set of S) defined by A B A B {displaystyle Aleq Biff Asubseteq B} . We may also partially order P ( S ) {displaystyle {mathcal {P}}(S)} by reverse set inclusion by defining A B B A {displaystyle Aleq Biff Bsubseteq A} . When quantified, A ⊆ B is represented as: ∀x{x∈A → x∈B}. PROPERTY * A FINITE set A is a SUBSET of (or is included in) B, denoted by A B {displaystyle Asubseteq B} , if and only if the cardinality of their intersection is equal to the cardinality of A. Formally: A B A B = A {displaystyle Asubseteq BLeftrightarrow Acap B=A} ⊂ 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 ⊇. 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, instead of ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and Links: ------ /wiki/Mathematics |