We say a set $A$ is a subset of another set $B$ if every element of $A$ is also an element of $B$, and we denote this by $A \subset B$. The empty set is a subset of every set, and every set is a subset of itself. The notation $A \subseteq B$ emphasizes that $A$ may be equal to $B$, while $A \subsetneq B$ says that $A$ is any subset of $B$ other than $B$ itself. In the latter case, $A$ is called a proper subset.

The following is a true statement:

$\emptyset \subset \{1, 2\} \subset \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C} \subset \mathbb{C}\cup\{\textrm{Groucho, Harpo, Chico}\} \supset \{1, 2, i, \textrm{Groucho}\}$

The set of all subsets of a given set $S$ is called the power set of $S$ and is denoted $\mathcal{P}(S)$ or $2^S$. The number of subsets of $S$ is $2^{|S|}$.

Example Problems