Class 11 Maths Chapter 1 Sets – Exercise 1.5 NCERT Solutions

Introduction

Exercise 1.5 focuses on practical problems on sets. Students learn how to apply set operations to real‑life situations, including union, intersection, and complement of sets. This exercise strengthens problem‑solving skills and prepares students for probability and relations.

Key Concepts

1. Union of Sets:

$$A\cup B=\{x:x\in A\text{ or }x\in B\}$$

2. Intersection of Sets:

$$A\cap B=\{x:x\in A\text{ and }x\in B\}$$

3. Complement of a Set:

$$A^{\mathrm{\prime }}=\{x:x\mathrm{\notin }A\}$$

4. Cardinality Relation:

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

Common Mistakes

• Forgetting to subtract intersection when calculating union.

• Misinterpreting complement as difference.

• Confusing universal set with given set.

• Ignoring overlapping elements in Venn diagrams.

NCERT Questions with Step‑by‑Step Solutions (10)

Q1. In a group of 70 students, 30 like cricket, 20 like football, and 35 like hockey. 5 like all three. Find how many like at least one game.

$$n(C\cup F\cup H)=n(C)+n(F)+n(H)-n(C\cap F)-n(F\cap H)-n(C\cap H)+n(C\cap F\cap H)$$

Substitute values accordingly → solution.

Q2. In a survey of 100 people, 60 like tea, 50 like coffee, and 30 like both. Find how many like neither.

$$n(T\cup C)=n(T)+n(C)-n(T\cap C)=60+50-30=80$$

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

Q3. Out of 200 students, 120 study Maths, 80 study Physics, and 60 study both. Find how many study only Maths.

$$n(\text{only Maths})=n(M)-n(M\cap P)=120-60=60$$

Q4. In a class of 50 students, 25 study Hindi, 20 study English, and 10 study both. Find how many study neither.

$$n(H\cup E)=25+20-10=35$$

$$n(\text{neither})=50-35=15$$

Q5. In a group of 60 people, 25 like apples, 30 like bananas, and 10 like both. Find how many like only bananas.

$$n(\text{only Bananas})=n(B)-n(A\cap B)=30-10=20$$

Q6. In a survey of 80 people, 40 like music, 30 like dance, and 20 like both. Find how many like only music.

$$n(\text{only Music})=n(M)-n(M\cap D)=40-20=20$$

Q7. In a class of 100 students, 60 study Biology, 50 study Chemistry, and 30 study both. Find how many study only Chemistry.

$$n(\text{only Chemistry})=n(C)-n(B\cap C)=50-30=20$$

Q8. In a group of 90 students, 40 play football, 30 play basketball, and 20 play both. Find how many play at least one game.

$$n(F\cup B)=40+30-20=50$$

Q9. In a survey of 120 people, 70 like reading, 60 like writing, and 40 like both. Find how many like neither.

$$n(R\cup W)=70+60-40=90$$

$$n(\text{neither})=120-90=30$$

Q10. In a class of 60 students, 35 study Maths, 25 study Science, and 15 study both. Find how many study only Maths.

$$n(\text{only Maths})=35-15=20$$

FAQs (10)

FAQ1. What is union of sets? All elements belonging to either set.

FAQ2. What is intersection of sets? Common elements of sets.

FAQ3. What is complement of a set? Elements not in the set but in universal set.

FAQ4. What is cardinality relation? $n(A\cup B)=n(A)+n(B)-n(A\cap B)$.

FAQ5. Why subtract intersection in union? To avoid double counting.

FAQ6. What is universal set? Set containing all elements under consideration.

FAQ7. What is difference between complement and difference? Complement is relative to universal set, difference is between two sets.

FAQ8. What is Venn diagram? Graphical representation of sets.

FAQ9. Why is Exercise 1.5 important? It applies set theory to practical problems.

FAQ10. What is real‑life use of sets? Surveys, data analysis, probability.

Conclusion

Exercise 1.5 covers practical problems on sets with solved examples and FAQs. Mastering these problems helps students apply set theory to real‑life situations and prepares them for probability.

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