Subset

A set $A$ is a subset of $B$ when every element of $A$ belongs to $B$.

A set $B$ is a superset of $A$ when $A$ is a subset of $B$, meaning that every element of $A$ belongs to $B$. Equality is permitted; a proper superset additionally differs from $A$.

Set inclusion $C\subseteq D$ means that every set element of the set $C$ also belongs to the set $D$. Strict set inclusion additionally requires $C\ne D$. A maximal clique cannot grow through strict set inclusion, while a maximum clique maximizes cardinality.

To prove equality of two sets, show each set inclusion separately. If both inclusions hold, every element belongs to one set exactly when it belongs to the other; the axiom of extensionality gives equality. Each direction may use different hypotheses. For example, when $x\in A$, the condition $x\notin A\setminus B$ implies $x\in B$; without $x\in A$, that inference would be invalid.