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 $A\subseteq B$ .