Subset

In mathematics, a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> A is a subset of a set B if all <a href="/facts/Element_(mathematics)/Q2mx1vHL">elements</a> of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. 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 (or contains) A or A is included (or contained) in B. A k-subset is a subset with k elements.
When quantified, 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
 is represented as 
 
 
 
 ∀
 x
 
 (
 
 x
 ∈
 A
 ⇒
 x
 ∈
 B
 
 )
 
 .
 
 
 {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right).}

One can prove the statement 
 
 
 
 A
 ⊆
 B
 
 
 {\displaystyle A\subseteq B}
 
 by applying a proof technique known as the element argument:Let sets A and B be given. To prove that 
 
 
 
 A
 ⊆
 B
 ,
 
 
 {\displaystyle A\subseteq B,}

<ol><li>suppose that a is a particular but arbitrarily chosen element of A</li>
<li>show that a is an element of B.</li></ol>
The validity of this technique can be seen as a consequence of <a href="/facts/Universal_generalization/XmPqQljM">universal generalization</a>: the technique shows 
 
 
 
 (
 c
 ∈
 A
 )
 ⇒
 (
 c
 ∈
 B
 )
 
 
 {\displaystyle (c\in A)\Rightarrow (c\in B)}
 
 for an arbitrarily chosen element c. Universal generalisation then implies 
 
 
 
 ∀
 x
 
 (
 
 x
 ∈
 A
 ⇒
 x
 ∈
 B
 
 )
 
 ,
 
 
 {\displaystyle \forall x\left(x\in A\Rightarrow x\in B\right),}
 
 which is equivalent to 
 
 
 
 A
 ⊆
 B
 ,
 
 
 {\displaystyle A\subseteq B,}
 
 as stated above.

Subset open-in-new

Subset