Subset - wikipedia Subset `` Superset '' redirects here. For other uses, see Superset ( disambiguation ). In mathematics, 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. A is a subset of B may also be expressed as B includes A ; or A is included in B. 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. Contents 1 Definitions 2 Properties 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 B, denoted by A ⊆ B, ( displaystyle A subseteq B, ) or equivalently B is a superset of A, denoted by B ⊇ A. ( displaystyle 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 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 A subsetneq B. ) or equivalently B is a proper superset of A ; this is written as B ⊋ A. ( displaystyle B supsetneq 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 A leq B iff A subseteq B ). We may also partially order P ( S ) ( displaystyle ( mathcal ( P ) ) ( S ) ) by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A. ( displaystyle A leq B iff B subseteq A. ) When quantified, A ⊆ B is represented as : ∀ x ( x ∈ A → x ∈ B ). Properties A set A is a subset of B if and only if their intersection is equal to A. Formally : A ⊆ B ⇔ A ∩ B = A. ( displaystyle A subseteq B Leftrightarrow A cap B = A. ) A set A is a subset of B if and only if their union is equal to B. Formally : A ⊆ B ⇔ A ∪ B = B. ( displaystyle A subseteq B Leftrightarrow A cup B = B. ) A finite set A is a subset of B if and only if the cardinality of their intersection is equal to the cardinality of A. Formally : A ⊆ B ⇔ A ∩ B = A. ( displaystyle A subseteq B Leftrightarrow A cap B = A. ) ⊂ And ⊃ symbols Euler diagram showing A is a proper subset of B, A ⊂ B, and conversely B is a proper superset of A. 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 proper superset respectively ; that is, with the same meaning and instead of the symbols, ⊊ and ⊋. This usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, but if A ⊂ B, then A definitely does not equal B. Examples The regular polygons form a subset of the polygons The set A = ( 1, 2 ) is a proper subset of B = ( 1, 2, 3 ), thus both expressions A ⊆ B and A ⊊ B are true. The set D = ( 1, 2, 3 ) is a subset of E = ( 1, 2, 3 ), thus D ⊆ E is true, and D ⊊ E is not true ( false ). Any set is a subset of itself, but not a proper subset. ( X ⊆ X is true, and X ⊊ X is false for any set X. ) The empty set ( ), denoted by ∅, is also a subset of any given set X. It is also always a proper subset of any set except itself. The set ( x : x is a prime number greater than 10 ) is a proper subset of ( x : x is an odd number greater than 10 ) The set of natural numbers is a proper subset of the set of rational numbers ; likewise, the set of points in a line segment is a proper subset of the set of points in a line. These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality ( the concept that corresponds to size, that is, the number of elements, of a finite set ) as the whole ; such cases can run counter to one 's initial intuition. The set of rational numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the latter set has a larger cardinality ( or power ) than the former set. Another example in an Euler diagram : A is a proper subset of B C is a subset but not a proper subset of B Other properties of inclusion A ⊆ B and B ⊆ C imply A ⊆ C Inclusion is the canonical partial order in the sense that every partially ordered set ( X, ⪯ ( displaystyle preceq ) ) 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 P ( S ) ( displaystyle ( mathcal ( P ) ) ( S ) ) 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, s,..., s ) and associating with each subset T ⊆ S ( which is to say with each element of 2 ) the k - tuple from ( 0, 1 ) of which the ith coordinate is 1 if and only if s is a member of T. See also Containment order References Jump up ^ Rosen, Kenneth H. ( 2012 ). Discrete Mathematics and Its Applications ( 7th ed. ). New York : McGraw - Hill. p. 119. ISBN 978 - 0 - 07 - 338309 - 5. Jump up ^ Rudin, Walter ( 1987 ), Real and complex analysis ( 3rd ed. ), New York : McGraw - Hill, p. 6, ISBN 978 - 0 - 07 - 054234 - 1, MR 0924157 Jump up ^ Subsets and Proper Subsets ( PDF ), retrieved 2012 - 09 - 07 Jech, Thomas ( 2002 ). Set Theory. Springer - Verlag. ISBN 3 - 540 - 44085 - 2. External links Media related to Subsets at Wikimedia Commons Weisstein, Eric W. `` Subset ''. MathWorld. 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Question: what is the symbol of subset in math
Answer: ⊂