Complement of a Set - Everything Outside the Set

The complement of a set A is the set of all elements in the universal set that are not in A. It is the mathematical way of describing everything that a set leaves out. Complement is closely linked to the idea of “not” in logic and “1 − P(A)” in probability.

The laws connecting complement to union and intersection are named after the British mathematician and logician Augustus De Morgan, who stated them formally in his 1847 book Formal Logic. De Morgan was also a close friend and collaborator of George Boole, whose 1854 work An Investigation of the Laws of Thought turned “not”, “and”, and “or” into precise algebraic operations – the same operations that, a century later, became the logic gates (NOT, AND, OR) inside every computer processor.

Complement appears constantly in everyday reasoning too: if a weather forecast gives a 30% chance of rain, the complement – a 70% chance of no rain – is exactly |ξ| − |A| expressed as a probability instead of a count.

Notation and Definition

The complement of A is written A′ (read as “A prime” or “A complement”).
Sometimes written as A^c or Â depending on the textbook.
A′ = {x : x ∈ ξ and x ∉ A}
In a Venn diagram, A′ is everything outside the circle for A.

Key Properties of Complement

Property	Statement	Meaning
Complement law	A ∪ A′ = ξ	A set and its complement together make the universal set
Complement law	A ∩ A′ = ∅	A set and its complement share no elements
Double complement	(A′)′ = A	The complement of the complement returns the original set
Universal complement	ξ′ = ∅	The complement of the universal set is empty
Empty complement	∅′ = ξ	The complement of the empty set is the universal set

Cardinality of the Complement

|A′| = |ξ| − |A|
The complement has as many elements as the universal set minus the elements in A.

De Morgan's Laws

These two powerful laws connect complement with union and intersection:
(A ∪ B)′ = A′ ∩ B′ — the complement of a union is the intersection of the complements.
(A ∩ B)′ = A′ ∪ B′ — the complement of an intersection is the union of the complements.

Worked Examples

ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8, 10}. Find A′ and |A′|.

A′ = elements in ξ not in A = {1, 3, 5, 7, 9}. |A′| = 10 − 5 = 5.

ξ = {integers 1 to 20}. B = {multiples of 3}. Find B′.

B = {3, 6, 9, 12, 15, 18}. |B| = 6.
B′ = {1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20}. |B′| = 20 − 6 = 14.

Verify De Morgan's first law with A = {1,2,3} and B = {3,4,5}, ξ = {1,2,3,4,5,6}.

A ∪ B = {1,2,3,4,5}. (A ∪ B)′ = {6}.
A′ = {4,5,6}. B′ = {1,2,6}. A′ ∩ B′ = {6}. ✓
Both give {6}, confirming De Morgan’s first law.

Complement in Probability

In probability, the complement of event A is the event that A does not occur. P(A′) = 1 − P(A). This mirrors exactly the set complement rule |A′| = |ξ| − |A|.

Key Takeaways

A′ = all elements of ξ that are not in A.
|A′| = |ξ| − |A|.
A ∪ A′ = ξ and A ∩ A′ = ∅ – a set and its complement are always disjoint and together form the universal set.
De Morgan’s laws: (A ∪ B)′ = A′ ∩ B′ and (A ∩ B)′ = A′ ∪ B′.

Practice: Complement Cardinality

Cardinality of a Complement