A [[Set|set]] $A$ is a **subset** of a set $B$ if $x \in A$ implies $x \in B$, and we write $A \subset B$ That is, all [[Element of a Set|elements]] of $A$ are also [[Element of a Set|elements]] of $B$. Sometimes written in reverse, so that $B$ is the [[superset]] of $A$ $B \supset A$ A set $A$ is a **proper subset** of $B$ if $A \subset B$ and $A\neq B$. To combine these two statements, we write $A \subsetneq B$ A set $A$ is an **improper subset** of $B$ if $A \subset B$, but the [[Equality of sets|set equality]] is still possible. $A \subseteq B$