Class 11 Maths Chapter 1 Sets – Exercise 1.3 NCERT Solutions

Introduction

Exercise 1.3 focuses on Venn diagrams and set identities. You will learn how to represent sets graphically, verify identities, and solve problems using union, intersection, difference, and complement. These concepts are widely applied in logic, probability, and computer science.

Formula Used

1. Union:

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

2. Intersection:

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

3. Difference:

$$A-B=\{x:x\in A\text{ and }x\mathrm{\notin }B\}$$

4. Complement:

$$A^{\mathrm{\prime }}=\{x:x\in U\text{ and }x\mathrm{\notin }A\}$$

5. Key Identities:

   • $A\cup A^{\mathrm{\prime }}=U$

   • $A\cap A^{\mathrm{\prime }}=\mathrm{\varnothing }$

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

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

NCERT Questions with Solutions (10)

Q1. Verify $A\cup A^{\mathrm{\prime }}=U$. Since every element is either in $A$ or not in $A$,

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

Q2. Verify $A\cap A^{\mathrm{\prime }}=\mathrm{\varnothing }$. No element can be both in $A$ and not in $A$.

$$A\cap A^{\mathrm{\prime }}=\mathrm{\varnothing }$$

Q3. Verify De Morgan’s law: $(A\cup B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}$. Using Venn diagram, shaded regions match.

Q4. Verify De Morgan’s law: $(A\cap B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cup B^{\mathrm{\prime }}$. Using Venn diagram, shaded regions match.

Q5. If $U=\{1,2,3,4,5,6\},A=\{1,2,3\},B=\{3,4,5\}$, find $A\cup B$.

$$A\cup B=\{1,2,3,4,5\}$$

Q6. With same sets, find $A\cap B$.

$$A\cap B=\{3\}$$

Q7. With same sets, find $A-B$.

$$A-B=\{1,2\}$$

Q8. With same sets, find $B-A$.

$$B-A=\{4,5\}$$

Q9. With same sets, find $A^{\mathrm{\prime }}$.

$$A^{\mathrm{\prime }}=\{4,5,6\}$$

Q10. With same sets, find $B^{\mathrm{\prime }}$.

$$B^{\mathrm{\prime }}=\{1,2,6\}$$

FAQs (10)

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

2. What is union in Venn diagram? Shaded region covering both sets.

3. What is intersection in Venn diagram? Shaded region common to both sets.

4. What is complement in Venn diagram? Region outside the set but within universal set.

5. What is difference in Venn diagram? Region in one set excluding the other.

6. What is De Morgan’s law? $(A\cup B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cap B^{\mathrm{\prime }}$, $(A\cap B{)}^{\mathrm{\prime }}=A^{\mathrm{\prime }}\cup B^{\mathrm{\prime }}$.

7. What is universal set in Venn diagram? Rectangle representing all elements under consideration.

8. What is empty set in Venn diagram? No shaded region.

9. Why use Venn diagrams? Easy visualization of set operations.

10. Why is this exercise important? It builds foundation for logic, probability, and advanced mathematics.

Conclusion

Exercise 1.3 has 10 solved questions and 10 FAQs that strengthen your understanding of Venn diagrams and set identities. This builds the foundation for advanced set theory in Class 11 Maths.

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