Set Types

Empty Set

A set with no elements. Denoted by $\varnothing$.

Singleton Set

A set with one element. A common error is to confuse the empty set $\varnothing$ with the set $\{\varnothing \}$, which is a singleton set. $\varnothing \ne \{\varnothing \}$.

Subset

We say that $A$ is a subset of $B$ or that $A$ is contained in $B$ if every element of $A$ is also an element of $B$, in short if $x\in A$ then $x\in B$.

Notation

$A\subseteq B$, if not, $A⊈B$

If $A$ is a subset of $B$, and $A\ne B$, we call $A$ the proper subset of $B$ and write it as $A\subset B$.

Power Set

If $S$ is a set, then the set of all subsets of $S$ is called the power set of $S$, denoted by $\mathcal{P}(S)$.

Example

$A=\{1,2,3\}$

$\therefore \mathcal{P}(A)=\{\varnothing ,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$

$S=\{\varnothing \}$

$\therefore \mathcal{P}(S)=\{\varnothing ,\{\varnothing \}\}$

The cardinality of a power set is represented by the formula $|\mathcal{P}(X)|=2^n$.

Example

$X=\{y,z\}$

$|\mathcal{P}(X)|=2^n=2^2=4$

$\mathcal{P}(X)=\{\varnothing ,\{y\},\{z\},\{y,z\}\}$