 Set difference

The set difference $A\setminus B$ keeps the set elements of $A$ that do not belong to $B$:Thus set difference combines logical conjunction (AND) with logical negation (NOT): pass the first condition and fail the second. The definition applies to any two sets; $B$ need not be a subset of $A$.

For $A=\{1,2,3\}$ and $B=\{3,4\}$, the two orders give different answers:The common member $3$ is removed in both directions. A database filter illustrates the usefulness: if $A$ is the set of records matching the first condition and $B$ the set matching the second, $A\setminus B$ selects records matching the first but not the second. The Venn diagram in set union shades both directions separately.

The name relative complement emphasizes that the ambient set is $A$: $A\setminus B$ is the complement of $A\cap B$ within $A$. If a larger universal set $U$ contains $A$ and $B$, then $A\setminus B=A\cap (U\setminus B)$. Changing that larger universal set does not change the set difference, because membership still requires belonging to $A$. In ZF the set difference exists by the axiom schema of separation. It also gives relative complements within a fixed tuple domain in finite relation closure for set-theoretic coding.