# Relative complement

The rel­a­tive com­ple­ment of two sets $$A$$ and $$B$$, de­noted $$A \setminus B$$, is the set of el­e­ments that are in $$A$$ while not in $$B$$.

For­mally stated, where $$C = A \setminus B$$

$$x \in C \leftrightarrow (x \in A \land x \notin B)$$

That is, Iff $$x$$ is in the rel­a­tive com­ple­ment $$C$$, then $$x$$ is in $$A$$ and x is not in $$B$$.

For ex­am­ple,

• $$\{1,2,3\} \setminus \{2\} = \{1,3\}$$

• $$\{1,2,3\} \setminus \{9\} = \{1,2,3\}$$

• $$\{1,2\} \setminus \{1,2,3,4\} = \{\}$$

If we name the set $$U$$ as the set of all things, then we can define the Ab­solute com­ple­ment of the set $$A$$, $$A^\complement$$, as $$U \setminus A$$