Absolute complement

In set theory, the complement of a setA, often denoted by Ac{\displaystyle A^{c}} (or A′), is the set of elements not in A. When all elements in the universe, i.e. all elements under consideration, are considered to be members of a given set U, the absolute complement of A is the set of elements in U that are not in A. The relative complement of A with respect to a set B, also termed the set difference of B and A, written B∖A,{\displaystyle B\setminus A,} is the set of elements in B that are not in A. Absolute complementDefinitionIf A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the relative complement of A in U: Ac=U∖A={x∈U:x∉A}.{\displaystyle A^{c}=U\setminus A=\{x\in U:x otin A\}.}The absolute complement of A is usually denoted by Ac{\displaystyle A^{c}}. Other notations include A¯{\displaystyle {\overline {A}}}, A′,{\displaystyle A',}