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Concept
Complement (set theory)
Summary
In set theory, the complement of a set A, often denoted by A∁ (or A′), is the set of elements not in A.
When all sets in the universe, i.e. all sets 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.
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:
Or formally:
The absolute complement of A is usually denoted by A∁. Other notations include
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.
Let A and B be two sets in a universe U. The following identities capture important properties of absolute complements:
De Morgan's laws:
Complement laws:
(this follows from the equivalence of a conditional with its contrapositive).
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