Set complement

In set theory, a complement is a set containing things that are not members of the set. Complements come in two forms: An absolute complement is a set containing everything that is not a member of the set. A relative complement, on the other hand, is the difference between a two sets where the elements of one set are removed from another.

Absolute complements

An absolute complement is a set which contains all things that are not a member of the set. When used practically, the absolute complement is the set of all other things within the domain of discourse, in the same way that the universal quantifier, ∀, is taken to quantify over a domain.

The symbol ∁ is used to indicate a complement. So for the set S, ∁(S) is its complement.

Thus, if we consider the domain to be natural numbers, and S = { 2, 3 }, then S(Γ) is the set of all natural numbers except 1, 2 and 3. So, ∁(S) = { 1, 4, 5 … }.

An absolute complement may be defined as ∁(S) = {x|x∉S}, which states that the complement of S is the set of every x such that x is not an element of S.

With an absolute domain, in which the domain is all things, including all sets, a contradiction can be derived. This problem is known as Russell's Paradox.

Relative complements

A relative complement limits its scope to a comparison between two sets. A relative complement of the set S compared to the set T, for example, is a set of all elements of T that are not also elements of S. This is expressed by the notation T - S, or sometimes T \ S.

For example, if S = { 1, 2, 3 } and T = { 1, 2, 3, 4 }, then T - S = { 4 }.

A relative complement may be defined as T - S = {x|x∈T ∧ x∉S}, which states that the complement of S is the set of every x such that x is not an element of S.