Complement of a Set – Definition, Properties, Examples, Facts, FAQs

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What Is the Complement of a Set?

In set theory, the complement of a set is the set of all elements that belong to the universal set but not to the original set.

The complement of a set A is denoted by A’ or Ac.

Example:

Universal set $= U =$ Set of all integers 

$U = \left\{…, -\;3,\;-\;2,\;-\;1,\; 0,\; 1,\; 2,\; 3,…\right\}$

Let A be the set of even integers.

$A = \left\{…,\;-\;2,\; 0,\; 2,\; 4,…\right\}$

Here, the complement of A is the set of odd integers.

Complement of set $A = A’ = \left\{…,\;-\;3,\; -\;1,\; 1,\; 3,…\right\}$

Definition of Complement of a Set

The difference between the universal set U and set A is called the complement of that set A.

If U is a universal set and A be any subset of U, then the complement of A is the set of all members of the universal set U, which are not the elements of A. 

Mathematically, the complement of a set A with respect to a universal set U is defined as 

$A’ = \left\{x \in U : x \in A\right\}$

The complement of A is the set of all elements x in U that are not in A.

Complement of a Set Symbol

The complement of the set A is denoted as A’ or Ac . The notation of the complement of a set uses an apostrophe (‘) or a superscript c after the name of the set. 

Also, as we know that the complement of A is the difference between Universal set and the set A, we can write

$A’ = U\;-\;A$

Complement of a Set: Venn Diagram

A universal set is typically represented using a rectangular box. Subsets of the universal set are generally represented by a circle. The complement of a set is the region of the universal set outside the set A.

The Venn diagram of the complement of set A is shown below. Here, the shaded portion in yellow shows the complement of set A.

Venn diagram of the complement of a set

If we have two sets that intersect each other, the complements of sets can be represented as follows:

Complement of set A and B

Cardinality of the Complement of a Set

Let the cardinality of the set A be n(A), the cardinality of the universal set U be n(U), then the cardinality of the complement of A, which is represented by n(A’) or |A’| is given by

$n(A’) = n(U) \;-\; n(A)$

$|A’| = |U| \;-\; |A|$

Properties of the Complement of a Set

Let’s discuss the properties of the complement of a set.

Complement Laws

  • The union of a set A and its complement A’ is equal to the universal set U.

$A \cup A’ = U$

  • The intersection of set A and A’ is the empty set

$A \cap A’ = ∅$

Law of Double Complementation

The complement of the complement of a set A is equal to A itself.

$(A’)’ = A$

Law of Empty Set and Universal Set

An empty set or null set (∅) is the complement of the universal set. The universal set is the complement of the empty set.

$∅’ = U$

$U’ = ∅$

De Morgan’s Laws

De Morgan’s laws are a set of two fundamental laws in a set theory that relate the complement of set operations. They are named after the mathematician Augustus De Morgan, a British mathematician .

  • The complement of the union of two sets A and B is equal to the intersection of their complements.

            $(A \cup B)’ = A’ \cap B’$

  • The complement of the intersection of two sets A and B is equal to the union of their complements.
    $(A \cap B)’ = A’ \cup B’$

How to Find the Complement of a Set

To find the complement of a set, follow the steps mentioned below:

Step 1: Write the elements of the universal set. Write down all the elements in the original set.

Step 2: Find the difference between U and A. In other words, identify all the elements in the Universal set that do not belong to the original set. 

Step 3: The elements identified in step 3 form the complement of the set.

Example: $U = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\right\}$

$A = \left\{1, 3, 5, 7, 9\right\}$

$A’ = U\;-\;A = \left\{ 2, 4, 6, 8, 10\right\}$

Facts about the Complement of a Set

  • The complement of an empty set is the universal set.
  • The complement of a universal set is the empty set.
  • The set and its complement are disjoint sets.
  • The complement of a set is unique.
  • The cardinality of a complement of a set is the difference between the cardinality of the universal set and the cardinality of the given set.

Conclusion

In this article, we have discussed the complement of a set, the notation, the Venn diagram, the properties of the complement of a set, and the method to find the complement. Let’s solve a few examples and practice problems.

Solved Examples on the Complement of a Set

  1. Let set $B = \left\{\text{Monday, Tuesday, Wednesday, Friday}\right\}$. Find the complement of B.

Solution: 

$B = \left\{\text{Monday, Tuesday, Wednesday, Friday}\right\}$.

Here, $U =$ Set of all days in a week

$U = \left\{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\right\}$

Let’s find the complement of B.

$B’ = U\;-\;B$

$B’ = \left\{x | x \text{is a day of the week and x is not in B}\right\}$

$B’ = \left\{\text{Thursday, Saturday, Sunday}\right\}$.

  1. Let $Y = \left\{x | x \text{is a positive even integer}\right\}$. Find the complement of Y, if U is the set of positive integers.

Solution: 

$Y = \left\{x | x \text{is a positive even integer}\right\}$

Here, U is the set of positive integers.

Therefore, $Y’ = \left\{x | x \text{is a positive integer and x is odd}\right\}$ 

$Y’ = \left\{1,\; 3,\; 5,\; 7,\; 9, …\right\}$.

  1. If $U = \left\{1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9,\; 10\right\}$ and $A = \left\{2,\; 3,\; 5,\; 7\right\}$. Find the complement of A.

Solution:

$U = \left\{1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9,\; 10\right\}$

$A = \left\{2,\; 3,\; 5,\; 7\right\}$

$\therefore A’ = U \;-\; A = \left\{1,\; 4,\; 6,\; 8,\; 9,\; 10\right\}$

  1. If $B = \left\{x: x \text{is a vowel in English alphabets}\right\}$, then find B’.

Solution:

$B = \left\{ x : x \text{is a vowel in English alphabets.}\right\}$

$B = \left\{a,\; e,\; i,\; o,\; u\right\}$

$B’ = \left\{\text{b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z}\right\}$

$B’= \left\{x: x \text{is not a vowel in English alphabets.}\right\}$

$B’ = \left\{x: x \text{is a consonant in English alphabets.}\right\}$

  1. Let $U = \left\{1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9\right\}$. If $A = \left\{1,\; 2,\; 3\right\}$ and $B = \left\{4,\; 5,\; 6\right\}$ and $C = \left\{7,\; 8,\; 9\right\}$, then find (A U B U C)’.

Solution:

$A = \left\{1,\; 2,\; 3\right\}$ and $B = \left\{4,\; 5,\; 6\right\}$ and $C = \left\{7,\; 8,\; 9\right\}$

$A U B U C = \left\{1,\; 2,\; 3,\; 4,\; 5,\; 6,\; 7,\; 8,\; 9\right\} = U$

$\therefore (A U B U C)’ = U’ = ∅$ 

  1. If $U = {1,\; 2,\; 3,\; 4,\; 5}$ and $C = \left\{1,\; 3,\; 5\right\}$ then find (C’)’?

Solution:

$(C’)’ = C$

$U = \left\{1,\; 2,\; 3,\; 4,\; 5\right\}$ and $C = \left\{1,\; 3,\; 5\right\}$

$C’ = \left\{2,\; 4\right\}$

Again taking the complement of C’, we get

$\therefore (C’)’ = \left\{2,\; 4\right\}’ = \left\{1,\;3,\;5\right\} = C$

Practice Problems on Complement of a Set

Complement of a Set - Definition, Properties, Examples, Facts, FAQs

Attend this quiz & Test your knowledge.

1

$A \cup A' =$

A
A'
U
CorrectIncorrect
Correct answer is: U
$A \cup A' = U$
2

If $U = \left\{a,\; b,\; c,\; d,\; e,\; f\right\}$ and $B = \left\{a,\; c,\; e\right\}$, what is the complement of B?

$\left\{a,\; b,\; d,\; f\right\}$
$\left\{a, b, c, d, e, f\right\}$
The empty set
$\left\{b, d, f\right\}$
CorrectIncorrect
Correct answer is: $\left\{b, d, f\right\}$
The complement of B consists of all elements in the universal set that are not in B. $B’ = \left\{b, d, f\right\}$.
3

$U’ =$

$\left\{0\right\}$
$∅$
$\left\{1\right\}$
U
CorrectIncorrect
Correct answer is: $∅$
$U’ = ∅$
4

If $U = \left\{a,\; b\right\}$ and the set $D = \left\{b\right\}$, what is the complement of D?

$\left\{a, b\right\}$
$\left\{a\right\}$
$\left\{b\right\}$
$∅$
CorrectIncorrect
Correct answer is: $\left\{a\right\}$
$U = \left\{a,\; b\right\}$
$D = \left\{b\right\}$
$D’ = U\;-\;D = \left\{a\right\}$
5

$(A \cup B)' =$

$A' \cup B'$
$A' \cap B'$
$A \cup B$
$A \cap B$
CorrectIncorrect
Correct answer is: $A' \cap B'$
By De Morgan’s law, $(A \cup B)'= A' \cap B'$

Frequently Asked Questions about the Complement of a Set

The meaning of a complement set is the set of all elements in the universal set that are not in the given set.

The symbol used to denote the complement of a set is usually an apostrophe (‘) or a superscript c, placed after the set symbol. For example, if A is a set, then its complement can be denoted by A’ or Ac.

The formula for the complement of a set A with respect to a universal set U is given byA’ $= U\;-\;A = \left\{x \in U | x \in A\right\}$

A set and its complement have no element in common.Thus, $A \;-\; A’ = A$

The complement of the empty set is the universal set.