The complement of a set is the set of all elements in the universal set that are not present in the given set. If the universal set is UUU and the given set is AAA, then the complement of AAA, denoted as A′A'A′ or AcA^cAc, is mathematically defined as:
A′={x∈U:x∉A}A'=\{x\in U:x\notin A\}A′={x∈U:x∈/A}
In other words, the complement of a set AAA is the difference between the universal set and the set AAA:
A′=U−AA'=U-AA′=U−A
For example, if the universal set UUU consists of all prime numbers up to 25, and A={2,3,5}A=\{2,3,5\}A={2,3,5}, the complement of AAA would include all prime numbers in UUU except those in AAA, such as {7,11,13,17,19,23}\{7,11,13,17,19,23\}{7,11,13,17,19,23}. The complement is often visually represented in Venn diagrams as the region outside the set AAA within the universal set. These definitions and explanations are consistent across common mathematical sources on set theory.
