Complement of a Set

In set theory, the complement of a set A is the collection of all elements that belong to the universal set U but do NOT belong to A. Think of it as everything that is 'left over' once you remove A from the universe of discourse. The complement captures the idea of negation in the language of sets: if A describes some property, then the complement of A describes everything that fails that property. The concept of a universal set is essential here. The universal set U is the agreed-upon 'universe' of all objects under consideration in a particular problem. For example, if you are studying integers, U might be the set of all integers; if you are studying students in a school, U might be every enrolled student. The complement is always defined relative to this chosen U, so the same set A can have different complements depending on what U is. Complements appear throughout mathematics and everyday reasoning. In probability, the complement rule states that P(A') = 1 − P(A), making it easy to calculate the probability of an event not occurring. In logic, complementation corresponds to negation. In digital electronics, NAND and NOR gates are built on complementation. Understanding complements is therefore a gateway to many advanced topics in mathematics and computer science.

Key takeaways
  • Define the Universal Set
  • Identify the Set A
  • Apply the Complement Definition
  • Check Key Properties
  • De Morgan's Laws

Define the Universal Set

Before finding a complement, always state the universal set U. Every element that could possibly appear in your problem must be in U. Example: if we are working with single-digit positive integers, then U = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Nothing outside this collection is considered.

Identify the Set A

Choose the set whose complement you want. Example: A = {2, 4, 6, 8} (the even single-digit positive integers). Every element of A must itself be an element of U; if something is in A but not in U, the problem is ill-defined.

Apply the Complement Definition

The complement A' (also written A^c or Ā) consists of exactly those elements of U that are not in A. Formally: A' = {x ∈ U : x ∉ A}. Go through U element by element: 1 ∉ A so 1 ∈ A'; 2 ∈ A so 2 ∉ A'; 3 ∉ A so 3 ∈ A'; and so on. Result: A' = {1, 3, 5, 7, 9}.

Check Key Properties

Four fundamental properties always hold: (1) A ∪ A' = U — the set and its complement together make the whole universe. (2) A ∩ A' = ∅ — they share no elements. (3) (A')' = A — the complement of the complement returns the original set. (4) U' = ∅ and ∅' = U — the universal set and the empty set are complements of each other.

De Morgan's Laws

Two powerful laws link complements with unions and intersections: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'. These let you push a complement 'inside' a compound set expression by swapping ∪ and ∩. They are used constantly in logic, probability, and circuit design.

The Key Equation

A′={x∈U:x∉A}A' = \{x \in U : x \notin A\}
  • A' — Complement of A (set): All elements of the universal set that do not belong to A; also written A^c or Ā.
  • U — Universal Set (set): The agreed-upon collection of all objects relevant to the problem at hand.
  • x — Element variable (element): A generic element being tested for membership.
  • \notin — Not an element of (relation): Denotes that x does not belong to the set A.

Worked Example

Problem. Let U = {a, b, c, d, e, f, g} and let B = {b, d, f}. Find B', verify that B ∪ B' = U and B ∩ B' = ∅, and then find (B')' .

B′={a,c,e,g},B∪B′=U,B∩B′=∅,(B′)′=BB' = \{a, c, e, g\},\quad B \cup B' = U,\quad B \cap B' = \emptyset,\quad (B')' = B

Answer. B' = {a, c, e, g}; B ∪ B' = U; B ∩ B' = ∅; (B')' = B

Figure
Diagram
UA2, 4, 6, 8A′1, 3, 57, 9U = {1,2,3,4,5,6,7,8,9}, A = {2,4,6,8}, A′ = {1,3,5,7,9}Shaded region = A′ (all of U outside circle A)
Origins · 1 minGeorg Cantor and how this was discovered
GC
Georg Cantor
1845–1918 · German

Georg Cantor founded modern set theory in the 1870s–1880s, introducing the rigorous language of sets, subsets, unions, intersections, and complements. His definition of a set as any well-defined collection of distinct objects gave mathematics the foundation for the complement concept. He also proved that different infinities have different sizes — a revolutionary idea that changed mathematics forever. Cantor's ideas were so radical that the leading mathematician of his time, Leopold Kronecker, called him a 'corrupter of youth.' Despite fierce opposition, Cantor's set theory eventually became the universal language of mathematics.

  • Union of Sets — The union and complement together satisfy A ∪ A' = U, making union essential to understanding what the complement completes.
  • Intersection of Sets — The intersection and complement satisfy A ∩ A' = ∅ and appear together in De Morgan's Laws.
  • Universal Set — The complement is defined relative to the universal set; without U, the complement cannot be determined.
  • Complement Rule in Probability — P(A') = 1 − P(A) is a direct application of the set complement, one of the most useful shortcuts in probability.
  • De Morgan's Laws — De Morgan's Laws describe exactly how complements interact with unions and intersections, generalising the complement concept.