Class 11 Maths Chapter 1 Sets – Exercise 1.6 NCERT Solutions

Introduction

Exercise 1.6 focuses on algebra of sets. Students learn how to apply set identities such as distributive laws, De Morgan’s laws, and verify relations using Venn diagrams. This exercise strengthens logical reasoning and prepares students for probability and relations.

Key Concepts

1. Distributive Laws:

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

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

2. De Morgan’s Laws:

$$(A\cup B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}$$

$$(A\cap B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cup B^{\mathrm{\prime }}$$

3. Idempotent Laws:

$$A\cup A=A,A\cap A=A$$

4. Complement Laws:

$$A\cup A^{\mathrm{\prime }}=U,A\cap A^{\mathrm{\prime }}=\mathrm{\varnothing }$$

Common Mistakes

• Forgetting to apply distributive property correctly.

• Misinterpreting complement relative to universal set.

• Confusing union with intersection in De Morgan’s laws.

• Ignoring boundary cases like empty set or universal set.

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

Q1. Verify: $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$. Using distributive law, both sides are equal. Verified.

Q2. Verify: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$. Using distributive law, both sides are equal. Verified.

Q3. Show that $(A\cup B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}$. By De Morgan’s law, verified.

Q4. Show that $(A\cap B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cup B^{\mathrm{\prime }}$. By De Morgan’s law, verified.

Q5. Prove: $A\cup (A\cap B)=A$. Since $A\cap B\subseteq A$, union with $A$ gives $A$. Verified.

Q6. Prove: $A\cap (A\cup B)=A$. Since $A\subseteq A\cup B$, intersection with $A$ gives $A$. Verified.

Q7. Show that $A\cup A^{\mathrm{\prime }}=U$. Union of set and its complement is universal set. Verified.

Q8. Show that $A\cap A^{\mathrm{\prime }}=\mathrm{\varnothing }$. Intersection of set and its complement is empty set. Verified.

Q9. Verify: $(A\cup B)\cup C=A\cup (B\cup C)$. Union is associative. Verified.

Q10. Verify: $(A\cap B)\cap C=A\cap (B\cap C)$. Intersection is associative. Verified.

FAQs (10)

FAQ1. What is distributive law in sets? $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$.

FAQ2. What are De Morgan’s laws? $(A\cup B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}$, $(A\cap B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cup B^{\mathrm{\prime }}$.

FAQ3. What is idempotent law? $A\cup A=A,A\cap A=A$.

FAQ4. What is complement law? $A\cup A^{\mathrm{\prime }}=U,A\cap A^{\mathrm{\prime }}=\mathrm{\varnothing }$.

FAQ5. What is associative law? $(A\cup B)\cup C=A\cup (B\cup C)$.

FAQ6. What is commutative law? $A\cup B=B\cup A,A\cap B=B\cap A$.

FAQ7. What is universal set? Set containing all elements under consideration.

FAQ8. What is empty set? Set with no elements.

FAQ9. Why are set identities important? They simplify complex set expressions.

FAQ10. What is practical use of set algebra? Used in probability, logic, and computer science.

Conclusion

Exercise 1.6 covers algebra of sets with solved examples and FAQs. Mastering these identities helps students in probability, relations, and logical reasoning.

visit:www.fuzymathacademy.com