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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,
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
Contents 1 Definitions 2 Properties 3 ⊂ and ⊃ symbols 4 Examples 5 Other properties of inclusion 6 See also 7 References 8 External links Definitions[edit] 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 Asubseteq B, or equivalently B is a superset of 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 .[1] Properties[edit] 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 Asubseteq BLeftrightarrow Acap 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 Asubseteq BLeftrightarrow Acup 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 Asubseteq BLeftrightarrow Acap B=A. ⊂ and ⊃ symbols[edit] 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 A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and proper superset instead of ⊊ and ⊋.[3] 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[edit] 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[edit] 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
Containment order References[edit] ^ Rosen, Kenneth H. (2012). Discrete
Mathematics
Jech, Thomas (2002). Set Theory. Springer-Verlag. ISBN 3-540-44085-2. External links[edit] Weisstein, Eric W. "Subset". MathWorld. v t e Mathematical logic General Formal language Formation rule Formal proof Formal semantics Well-formed formula Set Element Class Classical logic Axiom Rule of inference Relation Theorem Logical consequence Type theory Symbol Syntax Theory Systems Formal system Deductive system Axiomatic system Hilbert style systems Natural deduction Sequent calculus Traditional logic Proposition Inference Argument Validity Cogency Syllogism Square of opposition Venn diagram Propositional calculus Boolean logic Boolean functions Propositional calculus Propositional formula Logical connectives Truth tables Many-valued logic Predicate logic First-order Quantifiers Predicate Second-order Monadic predicate calculus Naive set theory Set Empty set Element Enumeration Extensionality Finite set Infinite set Subset Power set Countable set Uncountable set Recursive set Domain Codomain Image Map Function Relation Ordered pair Set theory Foundations of mathematics
Zermelo–Fraenkel set theory
Axiom
Model theory Model Interpretation Non-standard model Finite model theory Truth value Validity Proof theory Formal proof Deductive system Formal system Theorem Logical consequence Rule of inference Syntax Computability theory Recursion Recursive set Recursively enumerable set Decision problem Church–Turing thesis Computable function Primitive recursive function v t e Set theory Axioms Choice countable dependent Constructibility (V=L) Determinacy Extensionality Infinity Limitation of size Pairing Power set Regularity Union Martin's axiom
Axiom
replacement specification Operations Cartesian product Complement De Morgan's laws Disjoint union Intersection Power set Set difference Symmetric difference Union Concepts Methods Cardinality Cardinal number (large) Class Constructible universe Continuum hypothesis Diagonal argument Element ordered pair tuple Family Forcing One-to-one correspondence Ordinal number Transfinite induction Venn diagram Set types Countable Empty Finite (hereditarily) Fuzzy Infinite Recursive Subset · Superset Transitive Uncountable Universal Theories Alternative Axiomatic Naive Cantor's theorem Zermelo General Principia Mathematica New Foundations Zermelo–Fraenkel von Neumann–Bernays–Gödel Morse–Kelley Kripke–Platek Tarski–Grothendieck Paradoxes Problems Russell's paradox Suslin's problem Burali-Forti paradox Set theorists Abraham Fraenkel Bertrand Russell Ernst Zermelo Georg Cantor John von Neumann Kurt Gödel Paul Bernays Paul Cohen Richard Dedekind Thomas Jech Thoralf Sk |
