Class 11 Maths Chapter 1 Sets – Exercise 1.4 NCERT Solutions

Class 11 Maths Chapter 1 Sets – Exercise 1.4 NCERT Solutions

Introduction

Exercise 1.4 focuses on practical problems involving sets. You will learn how to apply union, intersection, and complement of sets to solve real‑life word problems such as students choosing subjects, survey data, and overlapping groups. This exercise strengthens your ability to interpret and solve problems using set theory.

Formula Used

1. Union of sets:

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

2. Intersection of sets:

$$n(A\cap B)=n(A)+n(B)-n(A\cup B)$$

3. Complement of set:

$$A^{\mathrm{\prime }}=U-A$$

4. General Principle of Inclusion–Exclusion (for three sets):

$$n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)$$

NCERT Questions with Solutions (10)

Q1. In a survey, 50 students like mathematics, 30 like physics, and 20 like both. Find number of students who like at least one subject.

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)=50+30-20=60$$

Q2. In a group of 100 students, 60 like English, 40 like Hindi, and 20 like both. Find number of students who like neither.

$$n(A\cup B)=60+40-20=80$$

$$n(\text{Neither})=100-80=20$$

Q3. In a class of 60 students, 25 play cricket, 20 play football, 10 play both. Find number of students who play neither.

$$n(A\cup B)=25+20-10=35$$

$$n(\text{Neither})=60-35=25$$

Q4. In a group of 80 students, 35 study maths, 30 study biology, 10 study both. Find number of students who study neither.

$$n(A\cup B)=35+30-10=55$$

$$n(\text{Neither})=80-55=25$$

Q5. In a survey of 200 people, 100 like tea, 80 like coffee, 40 like both. Find number of people who like only tea.

$$n(\text{Only Tea})=n(\text{Tea})-n(\text{Both})=100-40=60$$

Q6. In a survey of 200 people, 100 like tea, 80 like coffee, 40 like both. Find number of people who like only coffee.

$$n(\text{Only Coffee})=80-40=40$$

Q7. In a survey of 200 people, 100 like tea, 80 like coffee, 40 like both. Find number of people who like neither.

$$n(A\cup B)=100+80-40=140$$

$$n(\text{Neither})=200-140=60$$

Q8. In a class of 100 students, 60 study maths, 50 study physics, 30 study both. Find number of students who study only maths.

$$n(\text{Only Maths})=60-30=30$$

Q9. In a class of 100 students, 60 study maths, 50 study physics, 30 study both. Find number of students who study only physics.

$$n(\text{Only Physics})=50-30=20$$

Q10. In a class of 100 students, 60 study maths, 50 study physics, 30 study both. Find number of students who study neither.

$$n(A\cup B)=60+50-30=80$$

$$n(\text{Neither})=100-80=20$$

FAQs (10)

1. What is union in practical problems? Total students who like at least one subject.

2. What is intersection in practical problems? Students who like both subjects.

3. What is complement in practical problems? Students who like neither subject.

4. Formula for union of two sets?

$$n(A\cup B)=n(A)+n(B)-n(A\cap B)$$

5. Formula for union of three sets? Inclusion–Exclusion principle.

6. What is principle of inclusion–exclusion? Formula to calculate union of three sets.

7. Why use Venn diagrams in practical problems? Easy visualization of overlapping groups.

8. Can union be greater than total? No, union cannot exceed universal set.

9. Can intersection be empty? Yes, if no student likes both subjects.

10. Why is this exercise important? It builds skills in solving real‑life problems using set theory.

Conclusion

Exercise 1.4 has 10 solved questions and 10 FAQs that strengthen your understanding of practical problems involving sets. This completes the Sets chapter in Class 11 Maths.

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