Learning the concept of complements in set theory is essential in mathematics. It allows us to analyze and manipulate sets in various ways, leading to valuable insights and solutions to different mathematical problems, and solving Practice Problems on Complement of a Set is the best way to learn it. 

In this article we will learn the concept of complements of Set, exploring its properties, and methods of finding complements of sets and solve a set of Practice Problems on Complement of a Set.

What is a Set?

In mathematics, a set is a well-defined collection of distinct objects, considered as an entity in its own right. These objects can be anything: numbers, letters, symbols, or even other sets. The objects in a set are called its elements or members.

For example, if N is the set of natural numbers, and N = {1, 2, 3, 4, 5,….,∞}.

Complement of a Set

Complement of a set, denoted by A′, refers to all the elements that are not in the given set A, but which belong to the universal set U. Mathematically, the complement of set A is defined as:

A’ = {x ϵ U: x ∉ A}

Complement of set

Practice Problems on Complement of a Set – Solved

Following are some practice questions that explain the concept of Complement of a Set:

1: Find the complement of the set A = {1, 2, 3, 4, 5} within the universal set U = {1, 2, 3, 4, 5}.

Since set A already includes all elements of universal set U, its complement is an empty set {}.

2: Find the complement of the set B = {a, b, c} within the universal set U = {a, b, c, d, e}.

Complement of set B within the universal set U consists of all elements in U that are not in B. Therefore, the complement of B is {d, e}.

3: Find the complement of the set C = {red, blue} within the universal set U = {red, blue, green, yellow}.

Complement of set C within the universal set U consists of all elements in U that are not in C. Therefore, the complement of C is {green, yellow}.

4: Find the complement of the set D = {0, 1, 2, 3} within the universal set U = {0, 1, 2, 3, 4, 5}.

Complement of set D within the universal set U consists of all elements in U that are not in D. Therefore, the complement of D is {4, 5}.

5: Find the complement of the set E = {cat, dog} within the universal set U = {cat, dog, bird, fish}.

Complement of set E within the universal set U consists of all elements in U that are not in E. Therefore, the complement of E is {bird, fish}.

6: Find the complement of the set F = {5, 10, 15} within the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}.

Complement of set F within the universal set U consists of all elements in U that are not in F. Therefore, the complement of F is {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14}.

7: Find the complement of the set G = {x, y, z} within the universal set U = {w, x, y, z}.

Complement of set G within the universal set U consists of all elements in U that are not in G. Therefore, the complement of G is {w}.

8: Find the complement of the set H = {apple, banana} within the universal set U = {apple, banana, orange, pear}.

Complement of set H within the universal set U consists of all elements in U that are not in H. Therefore, the complement of H is {orange, pear}.

9: Find the complement of the set I = {100, 200, 300} within the universal set U = {0, 100, 200, 300, 400, 500}.

Complement of set I within the universal set U consists of all elements in U that are not in I. Therefore, the complement of I is {0, 400, 500}.

10: Find the complement of the set J = {Monday, Tuesday, Wednesday} within the universal set U = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.

Complement of set J within the universal set U consists of all elements in U that are not in J. Therefore, the complement of J is {Thursday, Friday, Saturday, Sunday}.

Practice Problems on Complement of a Set – IUnsolved

Q1. What is the complement of the set A = {1, 2, 3, 4} within the universal set U = {1, 2, 3, 4, 5, 6, 7}?

Q2. What is the complement of the set B = {a, b, c, d} within the universal set U = {a, b, c, d, e, f, g}?

Q3. What is the complement of the set C = {red, blue, green} within the universal set U = {red, blue, green, yellow, orange}?

Q4. What is the complement of the set D = {0, 1, 2} within the universal set U = {0, 1, 2, 3, 4, 5}?

Q5. What is the complement of the set E = {cat, dog, fish} within the universal set U = {cat, dog, bird, fish, rabbit}?

Q6. What is the complement of the set F = {5, 10, 15, 20} within the universal set U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}?

Q7. What is the complement of the set G = {x, y, z, w} within the universal set U = {w, x, y, z, a, b, c, d}?

Q8. What is the complement of the set H = {apple, banana, orange} within the universal set U = {apple, banana, orange, pear, mango}?

Q9. What is the complement of the set I = {100, 200, 300, 400} within the universal set U = {0, 100, 200, 300, 400, 500, 600, 700}?

Q10. What is the complement of the set J = {Monday, Tuesday, Wednesday, Thursday} within the universal set U = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}?

Practice Problems on Complement of a Set – FAQs

What is a complement of a set?

A complement of a set is a set containing all elements that are not in the original set but are present in the universal set.

How do you find the complement of a set?

To find the complement of a set, subtract the elements of the original set from the elements of the universal set.

Can a complement of a set be empty?

Yes, a complement of a set can be empty if the original set includes all elements of the universal set.

Is the complement of a set unique?

No, the complement of a set is not unique. It depends on the choice of the universal set.

What is the relationship between a set and its complement?

Complement of a set contains all elements that are not in the original set. In other words, it represents the “opposite” or “negation” of the original set within the universal set.