Subset

In set theory, a set $A$ is called a subset of a set $B$ if all of the elements of $A$ are contained in $B$. For example, any set is a subset of itself. Another example of a subset is a proper subset: a set $A$ is called a proper subset of a set $B$ if $A$ is subset of $B$ but is not equal to $B$.

A is a subset of B.

The symbol "$\subseteq$" always means "is a subset of." The symbol "$\subsetneq$" always means "is a proper subset of." There is also the symbol "$\subset$", which some authors use to mean "is a subset of" and other authors only use to mean "is a proper subset of."

For example:

• $\{3,7\}$ is a subset of $\{3,7\}$, so we could write $\{3,7\}\subseteq \{3,7\}$.

• $\{3,7\}$ is a proper subset of $\{1,3,4,7\}$, so we could write $\{3,7\}\subseteq \{1,3,4,7\}$,$\{3,7\}\subsetneq \{1,3,4,7\}$, or $\{3,7\}\subset \{1,3,4,7\}$.

• The interval [0, 1] is a proper subset of the set of real numbers $\mathbb {R}$, so $[0,1]\subset \mathbb {R}$.

Related pages

• Subset sum problem
• Vector subspace