Complement (set theory)

In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement.

Relative complement
If A and B are sets, then the relative complement of A in B, also known as the set-theoretic difference of B and A, is the set of elements in B, but not in A.

The relative complement of A in B is denoted B \ A (sometimes written B &minus; A, but this notation is ambiguous, as in some contexts it can be interpreted as the set of all b &minus; a, where b is taken from B and a from A).

Formally:
 * $$B \setminus A = \{ x\in B \, | \, x \notin A \}. $$

Examples:
 * {1,2,3} \ {2,3,4}  =   {1}
 * {2,3,4} \ {1,2,3}  =   {4}
 * If $$\mathbb{R}$$ is the set of real numbers and $$\mathbb{Q}$$ is the set of rational numbers, then $$ \mathbb{R}\setminus\mathbb{Q}$$ is the set of irrational numbers.

The following proposition lists some notable properties of relative complements in relation to the set-theoretic operations of union and intersection.

PROPOSITION 1: If A, B, and C are sets, then the following identities hold:
 * C \ (A ∩ B) =  (C \ A)∪(C \ B)
 * C \ (A ∪ B) =  (C \ A)∩(C \ B)
 * C \ (B \ A) =  (A ∩ C)∪(C \ B)
 * (B \ A) ∩ C =  (B ∩ C) \ A  =  B∩(C \ A)
 * (B \ A) ∪ C =  (B ∪ C) \ (A \ C)
 * A \ A =  Ø
 * Ø \ A =  Ø
 * A \ Ø =  A

Practical details
In the LaTeX typesetting language the command \setminus is usually used for rendering a set difference symbol – a backslash-like symbol. When rendered the \setminus command looks identical to \backslash except that it has a little more space in front and behind the slash, akin to the latex sequence \,\backslash\,.

The Mathematica programming language implements the operation with the Complement function.

The Matlab programming language implements the operation with the setdiff function.

Absolute complement
If a universe U is defined, then the relative complement of A in U is called the absolute complement (or simply complement) of A, and is denoted by AC (or sometimes A&prime;, also the same set often is denoted by $$\complement_U A$$ or $$\complement A$$ if U is fixed), that is:


 * AC = U \ A.

For example, if the universe is the set of natural numbers, then the complement of the set of odd numbers is the set of even numbers.

The following proposition lists some important properties of absolute complements in relation to the set-theoretic operations of union and intersection.

PROPOSITION 2: If A and B are subsets of a universe U, then the following identities hold:
 * De Morgan's laws:
 * (A ∪ B)C = AC ∩ BC
 * (A ∩ B)C = AC ∪ BC
 * Complement laws:
 * A ∪ AC =  U
 * A ∩ AC =  Ø
 * ØC =  U
 * UC =  Ø
 * If A⊆B, then BC⊆AC (this follows from the equivalence of a conditional with its contrapositive)
 * Involution or double complement law:
 * ACC =  A.
 * Relationships between relative and absolute complements:
 * A \ B = A ∩ BC
 * (A \ B)C = AC ∪ B

The first two complement laws above shows that if A is a non-empty subset of U, then {A, AC} is a partition of U.