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In set theory, the complement of a set A, often denoted by (or A′),[1] is the set of elements not in A.[2]
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 is the set of elements in B that are not in A.
Absolute complement
Definition
If 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:[3]
The absolute complement of A is usually denoted by . Other notations include [2] [4]
Examples
- Assume that the universe is the set of integers. If A is the set of odd numbers, then the complement of A is the set of even numbers. If B is the set of multiples of 3, then the complement of B is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
- Assume that the universe is the standard 52-card deck. If the set A is the suit of spades, then the complement of A is the union of the suits of clubs, diamonds, and hearts. If the set B is the union of the suits of clubs and diamonds, then the complement of B is the union of the suits of hearts and spades.
- When the universe is the universe of sets described in formalized set theory, the absolute complement of a set is generally not itself a set, but rather a proper class. For more info, see universal set.
Properties
Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:
Complement laws:[5]
-
- (this follows from the equivalence of a conditional with its contrapositive).
Involution or double complement law:
Relationships between relative and absolute complements:
Relationship with a set difference:
The first two complement laws above show that if A is a non-empty, proper subset of U, then {A, A∁} is a partition of U.
Relative complement
Definition
If A and B are sets, then the relative complement of A in B,[5] also termed the set difference of B and A,[6] is the set of elements in B but not in A.
The relative complement of A in B is denoted according to the ISO 31-11 standard. It is sometimes written but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements where b is taken from B and a from A.
Formally:
Examples
- If is the set of real numbers and is the set of rational numbers, then is the set of irrational numbers.
Properties
Let A, B, and C be three sets. The following identities capture notable properties of relative complements:
-
- with the important special case demonstrating that intersection can be expressed using only the relative complement operation.
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