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.
Notation and Definition
The complement of A is written A′ (read as “A prime” or “A complement”).
Sometimes written as Ac or  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
A′ = elements in ξ not in A = {1, 3, 5, 7, 9}. |A′| = 10 − 5 = 5.
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.
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 Questions
- ξ = {1 to 15}, A = {prime numbers}. Find A′ and |A′|.
- ξ = {letters a to j}, B = {a, e, i}. Find B′.
- If |ξ| = 40 and |A| = 17, find |A′|.
- Verify De Morgan’s second law with A = {2,4,6} and B = {4,6,8}, ξ = {1,2,3,4,5,6,7,8}.
- In probability terms: P(A) = 0.65. Find P(A′) and explain how this relates to the set complement.