Set difference

The set difference (or relative complement) of $B$ in $A$ is

Set difference formalizes “$A$ with the elements of $B$ removed.” In analysis it appears in punctured neighborhoods (e.g., $B(x,r)\setminus\{x\}$) and in decompositions like $(A\cup B)\setminus A = B\setminus A$.

Examples:

• $\{1,2,3\}\setminus\{2\}=\{1,3\}$.
• $(0,2)\setminus(1,3)=(0,1]$.
• For any set $A$, $A\setminus\varnothing = A$ and $A\setminus A=\varnothing$.