Subset

<h2 id="definition">Definition</h2>
If A and B are sets and every <a href="/facts/Element_(mathematics)/Q2mx1vHL">element</a> of A is also an element of B, then:

<ul><li>A is a subset of B, denoted by 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
, or equivalently,</li>
<li>B is a superset of A, denoted by 
 
 
 
 B
 ⊇
 A
 .
 
 
 {\displaystyle B\supseteq A.}
 
</li></ul>

If A is a subset of B, but A is not <a href="/facts/Equal_(math)/AQCuumLg">equal</a> to B (i.e. <a href="/facts/There_exists/hHveaBYo">there exists</a> at least one element of B which is not an element of A), then: 

<ul><li>A is a proper (or strict) subset of B, denoted by 
 
 
 
 A
 ⊊
 B
 
 
 {\displaystyle A\subsetneq B}
 
, or equivalently,</li>
<li>B is a proper (or strict) superset of A, denoted by 
 
 
 
 B
 ⊋
 A
 .
 
 
 {\displaystyle B\supsetneq A.}
 
</li></ul>
The <a href="/facts/Empty_set/ffEk3eA6">empty set</a>, written 
 
 
 
 {
 }
 
 
 {\displaystyle \{\}}
 
 or 
 
 
 
 ∅
 ,
 
 
 {\displaystyle \varnothing ,}
 
 has no elements, and therefore is <a href="/facts/Vacuous_truth/QBsue3pP">vacuously</a> a subset of any set X. 

<h2 id="basic-properties">Basic properties</h2>

<ul><li><a href="/facts/Reflexive_relation/NzTytEg9">Reflexivity</a>: Given any set 
 
 
 
 A
 
 
 {\displaystyle A}
 
, 
 
 
 
 A
 ⊆
 A
 
 
 {\displaystyle A\subseteq A}
 
<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li>
<li><a href="/facts/Transitive_relation/9r90y50r">Transitivity</a>: If 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
 and 
 
 
 
 B
 ⊆
 C
 
 
 {\displaystyle B\subseteq C}
 
, then 
 
 
 
 A
 ⊆
 C
 
 
 {\displaystyle A\subseteq C}
 
</li>
<li><a href="/facts/Antisymmetric_relation/YhuqG4z8">Antisymmetry</a>: If 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
 and 
 
 
 
 B
 ⊆
 A
 
 
 {\displaystyle B\subseteq A}
 
, then 
 
 
 
 A
 =
 B
 
 
 {\displaystyle A=B}
 
.</li></ul>
<h3>Proper subset</h3>
<ul><li><a href="/facts/Reflexive_relation/NzTytEg9">Irreflexivity</a>: Given any set 
 
 
 
 A
 
 
 {\displaystyle A}
 
, 
 
 
 
 A
 ⊊
 A
 
 
 {\displaystyle A\subsetneq A}
 
 is False.</li>
<li><a href="/facts/Transitive_relation/9r90y50r">Transitivity</a>: If 
 
 
 
 A
 ⊊
 B
 
 
 {\displaystyle A\subsetneq B}
 
 and 
 
 
 
 B
 ⊊
 C
 
 
 {\displaystyle B\subsetneq C}
 
, then 
 
 
 
 A
 ⊊
 C
 
 
 {\displaystyle A\subsetneq C}
 
</li>
<li><a href="/facts/Asymmetric_relation/rUdVRwuI">Asymmetry</a>: If 
 
 
 
 A
 ⊊
 B
 
 
 {\displaystyle A\subsetneq B}
 
 then 
 
 
 
 B
 ⊊
 A
 
 
 {\displaystyle B\subsetneq A}
 
 is False.</li></ul>
<h2 id="-and--symbols">⊂ and ⊃ symbols</h2>
Some authors use the symbols 
 
 
 
 ⊂
 
 
 {\displaystyle \subset }
 
 and 
 
 
 
 ⊃
 
 
 {\displaystyle \supset }
 
 to indicate subset and superset respectively; that is, with the same meaning as and instead of the symbols 
 
 
 
 ⊆
 
 
 {\displaystyle \subseteq }
 
 and 
 
 
 
 ⊇
 .
 
 
 {\displaystyle \supseteq .}
 
<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> For example, for these authors, it is true of every set A that 
 
 
 
 A
 ⊂
 A
 .
 
 
 {\displaystyle A\subset A.}
 
 (a <a href="/facts/Reflexive_relation/NzTytEg9">reflexive relation</a>).
Other authors prefer to use the symbols 
 
 
 
 ⊂
 
 
 {\displaystyle \subset }
 
 and 
 
 
 
 ⊃
 
 
 {\displaystyle \supset }
 
 to indicate proper (also called strict) subset and proper superset respectively; that is, with the same meaning as and instead of the symbols 
 
 
 
 ⊊
 
 
 {\displaystyle \subsetneq }
 
 and 
 
 
 
 ⊋
 .
 
 
 {\displaystyle \supsetneq .}
 
<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> This usage makes 
 
 
 
 ⊆
 
 
 {\displaystyle \subseteq }
 
 and 
 
 
 
 ⊂
 
 
 {\displaystyle \subset }
 
 analogous to the <a href="/facts/Inequality_(mathematics)/qkze4zdI">inequality</a> symbols 
 
 
 
 ≤
 
 
 {\displaystyle \leq }
 
 and 
 
 
 
 <
 .
 
 
 {\displaystyle <.}
 
 For example, if 
 
 
 
 x
 ≤
 y
 ,
 
 
 {\displaystyle x\leq y,}
 
 then x may or may not equal y, but if 
 
 
 
 x
 <
 y
 ,
 
 
 {\displaystyle x<y,}
 
 then x definitely does not equal y, and is less than y (an <a href="/facts/Irreflexive_relation/NzTytEg9">irreflexive relation</a>). Similarly, using the convention that 
 
 
 
 ⊂
 
 
 {\displaystyle \subset }
 
 is proper subset, if 
 
 
 
 A
 ⊆
 B
 ,
 
 
 {\displaystyle A\subseteq B,}
 
 then A may or may not equal B, but if 
 
 
 
 A
 ⊂
 B
 ,
 
 
 {\displaystyle A\subset B,}
 
 then A definitely does not equal B.

<h2 id="examples-of-subsets">Examples of subsets</h2>

<ul><li>The set A = {1, 2} is a proper subset of B = {1, 2, 3}, thus both expressions 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
 and 
 
 
 
 A
 ⊊
 B
 
 
 {\displaystyle A\subsetneq B}
 
 are true.</li>
<li>The set D = {1, 2, 3} is a subset (but not a proper subset) of E = {1, 2, 3}, thus 
 
 
 
 D
 ⊆
 E
 
 
 {\displaystyle D\subseteq E}
 
 is true, and 
 
 
 
 D
 ⊊
 E
 
 
 {\displaystyle D\subsetneq E}
 
 is not true (false).</li>
<li>The set {x: x is a <a href="/facts/Prime_number/L20FLkgn">prime number</a> greater than 10} is a proper subset of {x: x is an odd number greater than 10}</li>
<li>The set of <a href="/facts/Natural_number/u65og8JE">natural numbers</a> is a proper subset of the set of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a>; likewise, the set of points in a <a href="/facts/Line_segment/V8E0M4qk">line segment</a> is a proper subset of the set of points in a <a href="/facts/Line_(mathematics)/gJqsr4Gj">line</a>. These are two examples in which both the subset and the whole set are infinite, and the subset has the same <a href="/facts/Cardinality/m6VPojwT">cardinality</a> (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.</li>
<li>The set of <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> is a proper subset of the set of <a href="/facts/Real_number/R02gw5Pb">real numbers</a>. In this example, both sets are infinite, but the latter set has a larger cardinality (or power) than the former set.</li></ul>
Another example in an <a href="/facts/Euler_diagram/qBTWHJIV">Euler diagram</a>:

<h2 id="power-set">Power set</h2>
The set of all subsets of 
 
 
 
 S
 
 
 {\displaystyle S}
 
 is called its <a href="/facts/Power_set/FVuf8PDD">power set</a>, and is denoted by 
 
 
 
 
 
 P
 
 
 (
 S
 )
 
 
 {\displaystyle {\mathcal {P}}(S)}
 
.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>
The inclusion <a href="/facts/Binary_relation/r9BwGgaK">relation</a> 
 
 
 
 ⊆
 
 
 {\displaystyle \subseteq }
 
 is a <a href="/facts/Partial_order/qb4a3FAX">partial order</a> on the set 
 
 
 
 
 
 P
 
 
 (
 S
 )
 
 
 {\displaystyle {\mathcal {P}}(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
 
  if and only if 
 
 B
 ⊆
 A
 .
 
 
 {\displaystyle A\leq B{\text{ if and only if }}B\subseteq A.}

For the power set 
 
 
 
 
 
 P
 
 
 ⁡
 (
 S
 )
 
 
 {\displaystyle \operatorname {\mathcal {P}} (S)}
 
 of a set S, the inclusion partial order is—up to an <a href="/facts/Order_isomorphism/PwpVTjW5">order isomorphism</a>—the <a href="/facts/Cartesian_product/BfqBWzdL">Cartesian product</a> of 
 
 
 
 k
 =
 
 |
 
 S
 
 |
 
 
 
 {\displaystyle k=|S|}
 
 (the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of S) copies of the partial order on 
 
 
 
 {
 0
 ,
 1
 }
 
 
 {\displaystyle \{0,1\}}
 
 for which 
 
 
 
 0
 <
 1.
 
 
 {\displaystyle 0<1.}
 
 This can be illustrated by enumerating 
 
 
 
 S
 =
 
 {
 
 
 s
 
 1
 
 
 ,
 
 s
 
 2
 
 
 ,
 …
 ,
 
 s
 
 k
 
 
 
 }
 
 ,
 
 
 {\displaystyle S=\left\{s_{1},s_{2},\ldots ,s_{k}\right\},}
 
, and associating with each subset 
 
 
 
 T
 ⊆
 S
 
 
 {\displaystyle T\subseteq S}
 
 (i.e., each element of 
 
 
 
 
 2
 
 S
 
 
 
 
 {\displaystyle 2^{S}}
 
) the k-tuple from 
 
 
 
 {
 0
 ,
 1
 
 }
 
 k
 
 
 ,
 
 
 {\displaystyle \{0,1\}^{k},}
 
 of which the ith coordinate is 1 if and only if 
 
 
 
 
 s
 
 i
 
 
 
 
 {\displaystyle s_{i}}
 
 is a <a href="/facts/Set_membership/Q2mx1vHL">member</a> of T.
The set of all 
 
 
 
 k
 
 
 {\displaystyle k}
 
-subsets of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is denoted by 
 
 
 
 
 
 
 
 (
 
 
 A
 k
 
 
 )
 
 
 
 
 
 
 {\displaystyle {\tbinom {A}{k}}}
 
, in analogue with the notation for <a href="/facts/Binomial_coefficients/Tsx7eY5h">binomial coefficients</a>, which count the number of 
 
 
 
 k
 
 
 {\displaystyle k}
 
-subsets of an 
 
 
 
 n
 
 
 {\displaystyle n}
 
-element set. In <a href="/facts/Set_theory/1ptfsvUP">set theory</a>, the notation 
 
 
 
 [
 A
 
 ]
 
 k
 
 
 
 
 {\displaystyle [A]^{k}}
 
 is also common, especially when 
 
 
 
 k
 
 
 {\displaystyle k}
 
 is a <a href="/facts/Transfinite_number/m2oTjXaj">transfinite</a> <a href="/facts/Cardinal_number/ccoKwU4R">cardinal number</a>.

<h2 id="other-properties-of-inclusion">Other properties of inclusion</h2>
<ul><li>A set A is a subset of B <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> their intersection is equal to A. Formally:</li></ul>

A
        ⊆
        B
        
           if and only if 
        
        A
        ∩
        B
        =
        A
        .
      
    
    {\displaystyle A\subseteq B{\text{ if and only if }}A\cap B=A.}

<ul><li>A set A is a subset of B if and only if their union is equal to B. Formally:</li></ul>

A
        ⊆
        B
        
           if and only if 
        
        A
        ∪
        B
        =
        B
        .
      
    
    {\displaystyle A\subseteq B{\text{ if and only if }}A\cup B=B.}

<ul><li>A finite set A is a subset of B, if and only if the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> of their intersection is equal to the cardinality of A. Formally:</li></ul>

A
        ⊆
        B
        
           if and only if 
        
        
          |
        
        A
        ∩
        B
        
          |
        
        =
        
          |
        
        A
        
          |
        
        .
      
    
    {\displaystyle A\subseteq B{\text{ if and only if }}|A\cap B|=|A|.}

<ul><li>The subset relation defines a <a href="/facts/Partial_order/qb4a3FAX">partial order</a> on sets. In fact, the subsets of a given set form a <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebra</a> under the subset relation, in which the <a href="/facts/Join_and_meet/cYcLhRIR">join and meet</a> are given by <a href="/facts/Intersection_(set_theory)/RPfUNJh9">intersection</a> and <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a>, and the subset relation itself is the <a href="/facts/Inclusion_(Boolean_algebra)/z0CRxmwy">Boolean inclusion relation</a>.</li>
<li>Inclusion is the canonical <a href="/facts/Partial_order/qb4a3FAX">partial order</a>, in the sense that every partially ordered set 
 
 
 
 (
 X
 ,
 ⪯
 )
 
 
 {\displaystyle (X,\preceq )}
 
 is <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to some collection of sets ordered by inclusion. The <a href="/facts/Ordinal_number/GelFLjJt">ordinal numbers</a> are a simple example: if each ordinal n is identified with the set 
 
 
 
 [
 n
 ]
 
 
 {\displaystyle [n]}
 
 of all ordinals less than or equal to n, then 
 
 
 
 a
 ≤
 b
 
 
 {\displaystyle a\leq b}
 
 if and only if 
 
 
 
 [
 a
 ]
 ⊆
 [
 b
 ]
 .
 
 
 {\displaystyle [a]\subseteq [b].}
 
</li></ul>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Convex_subset/vdAuJRJl">Convex subset</a> – In geometry, set whose intersection with every line is a single line segmentPages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Inclusion_order/XDQqmtBz">Inclusion order</a> – Partial order that arises as the subset-inclusion relation on some collection of objects</li>
<li><a href="/facts/Mereology/WwzUjB95">Mereology</a> – Study of parts and the wholes they form</li>
<li><a href="/facts/Region_(mathematics)/sC90ftKL">Region</a> – Connected open subset of a topological spacePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Subset_sum_problem/Guvrwy0L">Subset sum problem</a> – Decision problem in computer science</li>
<li><a href="/facts/Hierarchy/mYJyqFrG">Subsumptive containment</a> – System of elements that are subordinated to each other</li>
<li><a href="/facts/Subspace_(mathematics)/G6rt4bQ8">Subspace</a> – Mathematical set with some added structurePages displaying short descriptions of redirect targets</li>
<li><a href="/facts/Total_subset/esqvbPDW">Total subset</a> – Subset T of a topological vector space X where the linear span of T is a dense subset of X</li></ul>

<h2 id="bibliography">Bibliography</h2>
<ul><li><a href="/facts/Thomas_Jech/E4JQu4Mk">Jech, Thomas</a> (2002). Set Theory. Springer-Verlag. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-44085-2.</li></ul>
<h2 id="external-links">External links</h2>
<ul><li> Media related to Subsets at Wikimedia Commons</li>
<li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Subset.html">"Subset"</a>. <a href="/facts/MathWorld/yc1jodbq">MathWorld</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Rosen, Kenneth H. (2012). Discrete Mathematics and Its Applications (7th ed.). New York: McGraw-Hill. p. 119. ISBN 978-0-07-338309-5. <a href="978-0-07-338309-5" target="_blank">978-0-07-338309-5</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Epp, Susanna S. (2011). Discrete Mathematics with Applications (Fourth ed.). Cengage Learning. p. 337. ISBN 978-0-495-39132-6. <a href="978-0-495-39132-6" target="_blank">978-0-495-39132-6</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Stoll, Robert R. (1963). Set Theory and Logic. San Francisco, CA: Dover Publications. ISBN 978-0-486-63829-4. <a href="978-0-486-63829-4" target="_blank">978-0-486-63829-4</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Rudin, Walter (1987), Real and complex analysis (3rd ed.), New York: McGraw-Hill, p. 6, ISBN 978-0-07-054234-1, MR 0924157 <a href="978-0-07-054234-1" target="_blank">978-0-07-054234-1</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Subsets and Proper Subsets (PDF), archived from the original (PDF) on 2013-01-23, retrieved 2012-09-07 <a href="https://web.archive.org/web/20130123202559/http://it.edgecombe.edu/homepage/killorant/MAT140/Module1/Subsets.pdf" target="_blank">https://web.archive.org/web/20130123202559/http://it.edgecombe.edu/homepage/killorant/MAT140/Module1/Subsets.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Weisstein, Eric W. "Subset". mathworld.wolfram.com. Retrieved 2020-08-23. <a href="https://mathworld.wolfram.com/Subset.html" target="_blank">https://mathworld.wolfram.com/Subset.html</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Subset open-in-new

Subset