Complement of a Set: Definition, Law, Formula, Properties, Examples

Complement of a set, Law of Complement, Property of Complement

Edited By Komal Miglani | Updated on Jul 02, 2025 06:39 PM IST

The complement of a set is a fundamental concept in set theory that helps identify elements not present in a given set, relative to a universal set. This operation is widely used in mathematics to simplify expressions, solve Venn diagram problems, and understand logical relationships between sets. If you want to select all records that do not meet a certain condition, you are dealing with the complement set. In computer science, a complement can represent the set of files not selected for backup. Understanding the definition, notation, and examples of the complement of a set is essential for students preparing for board exams and competitive tests. It also supports problem-solving in probability, algebra, and data classification. In this article, we will explain the meaning, formula, properties, and examples of the complement of a set in a clear and simple way.

This Story also Contains

Overview of Sets
Complement of a Set
Complement of Set: Venn Diagram
Laws of the Complement of a Set
Properties of Complement of Sets
Solved Examples Based on the Complement of Set
List of Topics Related to the Complement of a Set
NCERT Resources
Practice Questions on Complement of a Set

Complement of a set, Law of Complement, Property of Complement

Overview of Sets

Sets are a foundational concept in mathematics, central to various fields such as statistics, geometry, and algebra. A set is simply a collection of distinct objects, considered as a whole. These objects, called elements or members of the set, can be anything: numbers, people, letters, etc. Sets are particularly useful in defining and working with groups of objects that share common properties.

It is a well-defined collection of distinct objects and it is usually denoted by capital letters A, B, C, S, U,V...

Now, let us look into the definition that explain complement of set in detail.

Complement of a Set

The complement of a set is one of the fundamental operations in set theory. It is used to identify elements that do not belong to a specific set when compared with a larger reference set, known as the universal set. This concept is especially useful in problems involving Venn diagrams, logical operations, and data analysis. Before diving into the definition and examples of complement of a set, it's important to understand what sets and universal sets are.

Definition of Complement of a Set

Let $U$ be the universal set and $B$ be a subset of $U$. Then, the complement of set $B$ refers to the set of all elements in $U$ that are not in $B$.

Symbolic Representation: The complement of set $B$ is denoted by $B'$ or $B^C$ (read as "B complement").

$B' = \{x \in U \mid x \notin B\}$

This can also be written as: $B' = U - B$

In simple terms, the complement of a set B includes everything in the universal set except the elements of $B$.

Complement of a Set: Examples

Understanding the complement of a set with examples helps in applying the concept easily to solve problems based on universal sets and subsets. Below are a few solved examples that demonstrate how to find the complement of a set.

Example 1:
Let $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 4, 6, 7, 3, 8\}$
Then, the complement of set $A$ is:
$A' = \{2, 5, 9, 10\}$

Example 2:
Let $U = \{\text{blue, violet, green, yellow, grey, brown, black, white, red}\}$
Let $A = \{\text{violet, green, yellow, white}\}$ and $B = \{\text{grey, brown}\}$
Then:
$A' = \{\text{blue, grey, brown, black, red}\}$
$B' = \{\text{blue, violet, green, yellow, black, white, red}\}$

Example 3:
Let $U = \{x \in \mathbb{N} \mid x \leq 10\}$ and $A = \{x \in \mathbb{N} \mid x \leq 5\}$

Now,
$U = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ and $A = \{1, 2, 3, 4, 5\}$
So, $A' = \{6, 7, 8, 9\}$

These examples of complement of a set clearly show how to subtract a subset from its universal set to find the remaining elements.

Complement of Set: Venn Diagram

The Venn diagram for the complement of a set helps visualize all elements in the universal set that do not belong to the given set. In a typical Venn diagram, the universal set is represented by a rectangle, and the set in question is shown as a circle within it.

To illustrate the complement of set $A$, we shade the region outside the circle representing $A$, which includes all elements in $U$ that are not in $A$.

This visual representation makes it easier to understand how $A' = U - A$, and is widely used in logic problems and set theory applications.

Complement of a set

Laws of the Complement of a Set

The laws of complement of a set are key rules in set theory that describe the relationship between a set, its complement, the universal set, and the empty set. These properties are essential for simplifying set expressions and solving problems involving set operations and Venn diagrams.

Here are the fundamental complement laws in set theory:

Complementation Law:
The complement of the complement of a set returns the original set.
$\left(A'\right)' = A$
Universal Set Complement Law:
The complement of the universal set is the empty set.
$U' = \phi$
Empty Set Complement Law:
The complement of the empty set is the universal set. $\phi' = U$

Properties of Complement of Sets

The properties of complement of sets help in understanding how a set behaves with respect to its complement and the universal set. These properties are foundational in set theory, especially in solving problems using Venn diagrams, union, intersection, and De Morgan’s laws. Below are the main properties explained with examples.

1. Complement Laws

If $A$ is a subset of the universal set $U$, then $A'$ is also a subset of $U$.
The union of a set and its complement is the universal set: $A \cup A' = U$
The intersection of a set and its complement is the empty set: $A \cap A' = \emptyset$

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Example:
Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{4, 5\}$.
Then, $A' = \{1, 2, 3\}$
Now, $A \cup A' = \{1, 2, 3, 4, 5\} = U$
And, $A \cap A' = \emptyset$

2. Law of Double Complementation

The complement of the complement of a set gives the original set: $(A')' = A$
This means if $A'$ is the complement of $A$, then the complement of $A'$ brings us back to $A$.

Example:
Let $U = \{1, 2, 3, 4, 5\}$ and $A = \{4, 5\}$.
Then, $A' = \{1, 2, 3\}$ and $(A')' = \{4, 5\} = A$

3. Law of Empty Set and Universal Set

The complement of the universal set is the empty set: $U' = \emptyset$
The complement of the empty set is the universal set: $\emptyset' = U$

Example:
If $U = \{1, 2, 3, 4, 5\}$, then $U' = \emptyset$ and $\emptyset' = \{1, 2, 3, 4, 5\}$

4. De Morgan’s Laws

The complement of the union of two sets equals the intersection of their complements: $(A \cup B)' = A' \cap B'$
The complement of the intersection of two sets equals the union of their complements: $(A \cap B)' = A' \cup B'$

These are known as De Morgan’s Laws and are essential in logic and set theory.

Important Notes on the Complement of a Set

The complement of a set $A$ is denoted as $A'$ and is defined as: $A' = U - A$
A set and its complement are always disjoint.
The complement of the universal set is the empty set: $U' = \emptyset$
The complement of the empty set is the universal set: $\emptyset' = U$

These properties of set complement are frequently applied in mathematical logic, set operations, and Venn diagram-based questions.

Solved Examples Based on the Complement of Set

Example 1: Given $\mathrm{n}(\mathrm{U})=10, \mathrm{n}(\mathrm{A})=5, \mathrm{n}(\mathrm{B})=3$ and $n(A \cap B)=2$. A and B are subsets of $U$, then $n(A \cup B)^{\prime}=$

Solution:
Let $U$ be the universal set and $A$ a subset of $U$. Then the complement of $A$ is the set of all elements of $U$ which are not the elements of A. Symbolically, we write A' to denote the complement of $A$ with respect to $U$.
where $A^{\prime}=\{x: x \in U$ and $x \notin A\}$.Obviously $A^{\prime}=U-A$
$ \begin{aligned} & n(A \cup B)=5+3-2=6 \\ & n(A \cup B)^{\prime}=n(U)-n(A \cup B)=10-6=4 \end{aligned} $
Hence, the answer is 4 .

Example 2: Two newspapers A and B are published in a city. It is known that 25% of the city population reads A and 20% reads B while 8% reads both A and B. Further, 30% of those who read A but not B look into advertisements, and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. then the percentage of the population who look into advertisements is:

Solution:

Let $P(A)$ and $P(B)$ denote respectively the percentage of the city population that reads newspapers $A$ and $B$.

Let us consider the total percentage to be 100 . Then from the given data, we have

$P(A)=25, \quad P(B)=20, P(A \cap B)=8$

$\therefore$ Percentage of those who read $A$ but not $B$

$P(A \cap \bar{B})=P(A)-P(A \cap B)=25-8=17 \%$
And, Percentage of those who read $B$ but not $A$

$P(\bar{A} \cap B)=P(B)-P(A \cap B)=20-8=12 \%$

If $\mathrm{P}(\mathrm{C})$ denotes the percentage of those who look into an advertisement, then from the given data we obtain

$ \begin{aligned} & \therefore P(C)=30 \% \text { of } P(A \cap \bar{B})+40 \% \text { of } P(\bar{A} \cap B)+50 \% \text { of } P(A \cap B) \\ & \Rightarrow P(C)=\frac{3}{10} \times 17+\frac{2}{5} \times 12+\frac{1}{2} \times 8 \\ & \Rightarrow P(C)=13.9 \% \end{aligned} $
Hence, the answer is 13.9%.

Example 3: If $U=\{1,2,3,4,5\}, A=\{3,4,5\}$ and $B=\{1,2\}$. Then which of the following is true, if $U$ is a universal set of $A$ and $B$?
1) $A \subset B$
2) $A=B$
3) $A=B^{\prime}$
4) None of these

Solution:
Clearly B $=\mathrm{U}-\mathrm{A}$
Hence, $B=A^{\prime}$ and $A=B^{\prime}$
Hence, the answer is the option 3.

Example 4: If A and B are such sets that $A \cup B=U$ is the universal set. Which of the following must be true?
1) $A \cap B=\phi$
2) $A \cup B=A \cap B$
3) $A=B^c$
4) $A \cap U=A$

Solution:

$A \cup A^{\prime}=U$
A and B don't need to be compliment sets. It is only possible that
$A \cap U=A$
Hence, the answer is the option 4.

Example 5: If $A \cup B=U$ and $A \cap B=\phi$, then which of the following is not true?
1) $A^{\prime}=B$

2) $A=B^{\prime}$

3) $A \cap B=B \cap A$

4) $A \cup B=A \cap B$

Solution:
Clearly, $A$ and $B$ are complements of each other.
$\mathrm{A}=\mathrm{B}^{\prime}$ and $\mathrm{A}^{\prime}=\mathrm{B}$, so options (1) and (2) are correct.
Now option (3) is always correct as it is the commutative law.
In option (4), $A \cup B=U$ and $A \cap B=\phi$, so they are not equal.
Hence, the answer is the option 4.

List of Topics Related to the Complement of a Set

Explore key concepts that closely relate to the complement of a set, including foundational topics like set notations, subsets, and set operations. Understanding these topics strengthens your grasp of how sets interact within a universal set.

Roster And Set Builder Form Of Sets

Singleton Set

Equal And Equivalent Sets

Power Set

Subsets And Types Of Subsets

Difference Of Sets

Finite And Infinite Sets

Cardinal Number Of Some Sets

NCERT Resources

Explore essential NCERT resources for Class 11 Sets, including detailed solutions, clear revision notes, and handpicked exemplar problems. These materials are tailored to build a strong foundational understanding and support preparation for both board exams and competitive entrance tests.

NCERT Solutions for Class 11 Chapter 1 Sets

NCERT Notes for Class 11 Chapter 1 Sets

NCERT Exemplar for Class 11 Chapter 1 Sets

Practice Questions on Complement of a Set

Sharpen your understanding of the complement of a set with focused practice questions designed to reinforce key definitions, properties, and formulas. These practice MCQs are ideal for testing your conceptual clarity and preparing for board exams and entrance tests. Explore the links below to attempt topic-wise MCQs and advance your set theory skills systematically.

To practice questions based on Complement Of A Set - Practice Question MCQ, click here.

You can practice the next topics of Sets below:

Cardinal Number Of Some Sets - Practice Question MCQ

De Morgans Laws - Practice Question MCQ

Ordered Pair Cartesian Product Of Two Sets - Practice Question MCQ

Frequently Asked Questions (FAQs)

1. How many elements are there in complement of set A where $U = \{a,b,c,d,e,f,j,i,l \}$ and $A = \{a,l,i,j,c,e\}$?

The complement of set A is $\{b,d,f\}$. The number of elements in the complement of set A is $3$.

2. Define complement of set.

The complement of $A$ is the set of all elements of $U$ which are not the elements of $A$.

3. What is complementation law?

The complement of the complement of a set is the set itself.

4. What is universal complement law?

The complement of the universal set is the empty set.

5. Give the venn diagram for the complement of a set.

The venn diagram of complement of a set A and universal U is

6. What is the complement of a set?

The complement of a set A, usually denoted as A' or A^c, is the set of all elements in the universal set U that are not in A. In other words, it contains everything that is not in A but is within the context of the problem (universal set).

7. How do you visualize the complement of a set using a Venn diagram?

In a Venn diagram, the complement of a set A is represented by the area inside the universal set U but outside the circle representing set A. It's the shaded region that doesn't overlap with A.

8. What is the Law of Complement?

The Law of Complement states that for any set A, (A')' = A. This means that the complement of the complement of a set is the original set itself. It's like a double negative in language.

9. How does the Law of Complement relate to set operations?

The Law of Complement is fundamental in set theory as it allows us to simplify complex set expressions and proves that taking the complement twice brings us back to the original set. It's often used in conjunction with other set laws in proofs and problem-solving.

10. What is the relationship between a set and its complement in terms of the universal set?

A set and its complement are mutually exclusive (they have no elements in common) and collectively exhaustive (they cover the entire universal set). Mathematically, this is expressed as A ∪ A' = U and A ∩ A' = ∅.

11. Can the complement of a set be an empty set?

Yes, the complement of a set can be an empty set, but only if the original set is equal to the universal set. In this case, there are no elements in U that are not in the set, so its complement is empty.

12. What is the complement of the empty set?

The complement of the empty set is the universal set. Since the empty set contains no elements, its complement includes all elements in the universal set.

13. How does the size of a set relate to the size of its complement?

The size of a set and its complement are inversely related. If n(U) represents the number of elements in the universal set and n(A) represents the number of elements in set A, then n(A') = n(U) - n(A).

14. What is De Morgan's Law and how does it relate to complements?

De Morgan's Laws are two important rules in set theory that involve complements:

15. How can you use the concept of complement to solve probability problems?

The concept of complement is often used in probability to calculate the probability of an event not occurring. If P(A) is the probability of event A occurring, then P(A') = 1 - P(A) is the probability of A not occurring. This is especially useful when it's easier to calculate the probability of the complement event.

16. What is the difference between relative complement and absolute complement?

The absolute complement (or simply complement) of a set A is all elements in the universal set U that are not in A. The relative complement of A with respect to B (denoted B \ A or B - A) is the set of elements that are in B but not in A. The absolute complement is a special case of relative complement where B is the universal set.

17. Can a set be its own complement?

No, a set cannot be its own complement unless the universal set contains only two elements. In this special case, each set would be the complement of the other. In all other cases, a set and its complement are mutually exclusive and cannot be the same.

18. How does the concept of complement relate to subsets?

If A is a subset of B, then the complement of B is a subset of the complement of A. Mathematically, if A ⊆ B, then B' ⊆ A'. This relationship is known as the subset complement property.

19. What is the complement of the universal set?

The complement of the universal set is the empty set. Since the universal set contains all elements under consideration, there are no elements outside of it, making its complement empty.

20. How can you use the concept of complement to simplify set expressions?

The concept of complement can be used to simplify set expressions by applying laws like De Morgan's Laws, the Law of Complement, and other set identities. For example, (A ∪ B)' can be simplified to A' ∩ B' using De Morgan's Law.

21. What is the relationship between intersection and complement?

The intersection of a set and its complement is always the empty set. Mathematically, A ∩ A' = ∅. This is because no element can simultaneously be in a set and not in that set.

22. How does the concept of complement relate to set difference?

The set difference A - B (all elements in A that are not in B) can be expressed using complements as A ∩ B'. This shows that set difference and complement are closely related operations.

23. Can the complement of a set have more elements than the original set?

Yes, the complement of a set can have more elements than the original set. This occurs when the original set contains fewer than half of the elements in the universal set. The smaller the original set, the larger its complement.

24. How does the concept of complement apply to infinite sets?

The concept of complement applies to infinite sets in the same way as finite sets. The complement of an infinite set A is still all elements in the universal set U that are not in A. However, both A and A' can be infinite in this case.

25. What is the double complement property?

The double complement property, also known as the Law of Double Complement, states that (A')' = A. This means that taking the complement of a set twice results in the original set.

26. How can you use Venn diagrams to illustrate the properties of complements?

Venn diagrams are useful for visualizing complement properties. For example, you can use them to illustrate that A ∪ A' = U (the entire diagram is shaded) and A ∩ A' = ∅ (no overlap between A and its complement).

27. What is the relationship between union and complement?

The union of a set and its complement is always the universal set. Mathematically, A ∪ A' = U. This is because every element in the universal set must be either in A or not in A (in A').

28. How does the concept of complement relate to logical negation?

The complement of a set is analogous to logical negation in propositional logic. Just as "not A" in logic represents everything that isn't A, the complement A' in set theory represents all elements that aren't in A.

29. Can you have a complement of a complement of a complement?

Yes, you can take the complement of a set any number of times. However, due to the Law of Complement, (A')' = A, so (A')')'= A' . The result will always alternate between A and A' no matter how many times you take the complement.

30. How does the concept of complement apply in computer science, particularly in database queries?

In database queries, the concept of complement is often used in NOT operations. For example, to find all records NOT matching a certain condition, you're essentially looking for the complement of the set of records that do match the condition.

31. What is the relationship between complement and cardinality?

The cardinality of a set and its complement are related by the equation |A'| = |U| - |A|, where |A| denotes the cardinality (number of elements) of set A and U is the universal set. This relationship holds for both finite and infinite sets.

32. How can you use the concept of complement to prove set identities?

The concept of complement is often used in proving set identities. For example, to prove A = B, you could show that A' = B', or to prove A ⊆ B, you could show that B' ⊆ A'. This approach can sometimes simplify proofs.

33. What is the complement of an interval in real numbers?

For an interval [a,b] on the real number line, its complement is (-∞, a) ∪ (b, ∞), assuming the universal set is all real numbers. This represents all real numbers less than a or greater than b.

34. How does the concept of complement relate to the power set?

If P(A) is the power set of A (the set of all subsets of A), then for any subset B in P(A), its complement B' is also in P(A). This is because B' = A \ B, which is a subset of A.

35. What is the relationship between complement and symmetric difference?

The symmetric difference of sets A and B (denoted A Δ B) can be expressed using complements as (A ∪ B) \ (A ∩ B), or equivalently as (A \ B) ∪ (B \ A). This shows how complement (via set difference) is fundamental to defining symmetric difference.

36. How does the concept of complement apply in probability theory, particularly in the calculation of conditional probabilities?

In probability theory, the complement is often used in calculating conditional probabilities. For example, Bayes' theorem can be expressed using complements: P(A|B) = P(B|A)P(A) / [P(B|A)P(A) + P(B|A')P(A')], where A' is the complement of event A.

37. Can you have a partial complement of a set?

While not a standard term, a "partial complement" could refer to the relative complement. For sets A and B, the relative complement of A in B (B \ A) could be considered a partial complement of A, as it only considers elements in B rather than the entire universal set.

38. How does the concept of complement relate to the principle of inclusion-exclusion?

The principle of inclusion-exclusion uses complements implicitly. For example, the formula |A ∪ B| = |A| + |B| - |A ∩ B| can be derived using complements, as (A ∪ B)' = A' ∩ B'.

39. What is the relationship between complement and disjoint sets?

Two sets A and B are disjoint if and only if A is a subset of B'. In other words, sets are disjoint when each is contained entirely within the complement of the other.

40. How can you use the concept of complement to solve set theory problems involving three or more sets?

For problems involving multiple sets, complements can simplify expressions. For example, (A ∪ B ∪ C)' = A' ∩ B' ∩ C' by De Morgan's Law. This can make it easier to calculate or visualize complex set relationships.

41. What is the complement of a countable set in an uncountable universal set?

If A is a countable set and U is an uncountable universal set, then A' is uncountable. This is because the union of A (countable) and A' (uncountable) must equal U (uncountable), and the union of a countable set and a countable set is countable.

42. How does the concept of complement relate to the algebra of sets?

The complement operation is one of the fundamental operations in the algebra of sets, along with union and intersection. Many set algebraic laws and identities involve complements, such as De Morgan's Laws and the Law of Complement.

43. Can you have a complement of a multiset?

The concept of complement can be extended to multisets, but it's not as straightforward as with regular sets. One approach is to define the complement of a multiset A as a multiset where the multiplicity of each element is the difference between its multiplicity in the universal multiset and in A.

44. How does the concept of complement apply in fuzzy set theory?

In fuzzy set theory, the complement of a fuzzy set A is typically defined as a fuzzy set A' where the membership degree of each element x is 1 - μA(x), where μA(x) is the membership degree of x in A. This extends the classical set complement to fuzzy sets.

45. What is the relationship between complement and duality in set theory?

Complement plays a key role in the principle of duality in set theory. This principle states that for any theorem about sets, the dual theorem formed by interchanging ∪ and ∩, ⊆ and ⊇, and ∅ and U, is also true. Complement is central to this duality.

46. How can you use the concept of complement to simplify complex set expressions involving nested set operations?

For complex nested set expressions, applying complement properties can often simplify the expression. For example, ((A ∪ B)' ∩ C)' can be simplified to (A ∪ B) ∪ C' using De Morgan's Laws and the Law of Double Complement.

47. What is the relationship between complement and the concept of a partition of a set?

In a partition of a set S into subsets A1, A2, ..., An, each subset Ai is the complement of the union of all other subsets with respect to S. That is, Ai = S \ (A1 ∪ A2 ∪ ... ∪ Ai-1 ∪ Ai+1 ∪ ... ∪ An).

48. How does the concept of complement relate to the topology of sets?

In topology, the complement of a set A in a topological space X is X \ A. The properties of complements are important in defining closed sets (a set is closed if and only if its complement is open) and in various topological constructions and proofs.

49. Can you have a complement of an infinite union or intersection of sets?

Yes, the complement of an infinite union or intersection of sets is defined and follows De Morgan's Laws:

50. How does the concept of complement relate to the notion of a universe of discourse in logic and set theory?

The universe of discourse U in logic and set theory defines the context for complement operations. The complement of a set A is always taken with respect to U. Changing U can change the complement of A, highlighting the importance of clearly defining the universe of discourse in any problem.