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 $t\notin C$) nor $C\subseteq A$ (because $p\in C$ but $p\notin A$). The fact that $A$ is not a subset of $C$ is written as $A⊈C$. Similarly, we have $C⊈A$.

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 $A\ne B$. 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