Subset

If every element of set $A$ is also an element of set $B$, then $A$ is a subset of $B$, denoted $A\subseteq B$. Note that every set is a subset of itself, and the Empty Set is a subset of every set.

Context

Subsets are central to set theory and are used constantly in proofs. To prove $A\subseteq B$, you take an arbitrary element $x\in A$ and show $x\in B$. To prove set equality $A=B$, you typically prove $A\subseteq B$ and $B\subseteq A$.

See Also

• set
• proper subset
• power set
• Empty Set
• complement