# subset

Given two sets $A$ and $B$, we say that $A$ is a subset of $B$ (which we denote as $A\subseteq B$ or simply $A\subset B$) if every element of $A$ is also in $B$. That is, the following implication  holds:

 $x\in A\Rightarrow x\in B.$

The relation  between $A$ and $B$ is then called set inclusion.

Some examples:

The set $A=\{d,r,i,t,o\}$ is a subset of the set $B=\{p,e,d,r,i,t,o\}$ because every element of $A$ is also in $B$. That is, $A\subseteq B$.

On the other hand, if $C=\{p,e,d,r,o\}$, then neither $A\subseteq C$ (because $t\in A$ but $t\not\in C$) nor $C\subseteq A$ (because $p\in C$ but $p\not\in A$). The fact that $A$ is not a subset of $C$ is written as $A\not\subseteq C$. Similarly, we have $C\not\subseteq A$.

If $X\subseteq Y$ and $Y\subseteq X$, it must be the case that $X=Y$.

Every set is a subset of itself, and the empty set  is a subset of every other set. The set $A$ is called a proper subset   of $B$, if $A\subset B$ and $A\neq B$. In this case, we do not use $A\subseteq B$.

 Title subset Canonical name Subset Date of creation 2013-03-22 11:52:38 Last modified on 2013-03-22 11:52:38 Owner Wkbj79 (1863) Last modified by Wkbj79 (1863) Numerical id 13 Author Wkbj79 (1863) Entry type Definition Classification msc 03-00 Classification msc 00-02 Related topic EmptySet Related topic Superset  Related topic TotallyBounded Related topic ProofThatAllSubgroupsOfACyclicGroupAreCyclic Related topic Property2 Related topic CardinalityOfAFiniteSetIsUnique Related topic CriterionOfSurjectivity Defines set inclusion