The NCERT Solutions for Class 11 Maths Chapter 1 Sets Miscellaneous Exercise solve every question of the mixed exercise, according to the latest 2026-27 CBSE syllabus. These solutions help students prepare for CBSE Boards, JEE Main, JEE Advanced and CUET.
Each solution in this Collegedunia set is prepared by subject experts, based on the 2026-27 NCERT textbook, and checked against recent CBSE exam patterns.
The Miscellaneous Exercise pulls together every idea from the chapter. It has 10 questions built around these skills:
- Proving subset and set-equality results with clear reasoning.
- Applying union, intersection and difference in mixed problems.
- Using set identities and De Morgan's laws in proofs.
Topics Covered in Sets Miscellaneous Exercise
This exercise mixes every tool from Chapter 1. Students prove subset relations, show two sets are equal, and use the three operations together. Several questions ask for a short proof rather than a numerical answer, so clear reasoning matters as much as the final set.
- Subset proofs: showing that if A ⊂ B and B ⊂ C, then A ⊂ C.
- Set identities: results like A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A.
- Difference results: if A ⊂ B, then C − B ⊂ C − A.
- Counter-examples: showing A ∩ B = A ∩ C need not mean B = C.
Important: a proof needs both directions. To show two sets are equal, show each is a subset of the other.
Question-wise Breakdown of Sets Miscellaneous Exercise
The NCERT Solutions for Class 11 Maths Chapter 1 Sets Miscellaneous Exercise answer all 10 questions in order. The table shows what each question asks so students can plan their practice. Full step-by-step answers are inside the PDF above.
| Question | What it asks | Skill tested |
|---|---|---|
| Q1 | Decide which given sets are subsets of one another | Subsets |
| Q2 | State whether statements about subsets are true or false | Subset reasoning |
| Q3 | Check the transitive rule for subsets | Subset chain |
| Q4 | Prove C − B ⊂ C − A when A ⊂ B | Set difference proof |
| Q5–Q7 | Prove set identities using union, intersection and difference | Set identities |
| Q8 | Show A ∩ B = A ∩ C need not give B = C | Counter-example |
| Q9–Q10 | Prove equality of sets and build a required example | Mixed proofs |
Most questions ask for a proof, so the Miscellaneous Exercise is the best test of full-chapter understanding.
Key Formulas and Definitions for Sets Miscellaneous Exercise
The Miscellaneous Exercise uses results from every earlier exercise. Students should keep these key identities ready before attempting the proofs.
| Result | Meaning |
|---|---|
| A ⊂ B and B ⊂ C | gives A ⊂ C (transitive property) |
| A = (A ∩ B) ∪ (A − B) | splits A into shared and unshared parts |
| A ∪ (A ∩ B) | = A (absorption law) |
| A ∩ (A ∪ B) | = A (absorption law) |
| (A ∪ B)′ | = A′ ∩ B′ (De Morgan's law) |
| n(A ∪ B) | n(A) + n(B) − n(A ∩ B) |
Tip: to prove a set identity, take a general element and show it belongs to both sides. This element-chasing method solves Questions 5 to 7 cleanly.
Common Mistakes in Sets Miscellaneous Exercise
Students lose easy marks in the Miscellaneous Exercise by writing loose proofs. The list below covers the errors that show up most often on the NCERT Solutions for Class 11 Maths Chapter 1 Sets Miscellaneous Exercise.
- Proving equality in one direction only. Show each set is a subset of the other.
- Treating a single example as a proof. One case supports a claim but does not prove it.
- Skipping the reason in a true or false question. Always justify the answer with a rule or counter-example.
- Confusing A ∩ B = A ∩ C with B = C. The two need not be equal, as Question 8 shows.
Practice Questions for Sets Miscellaneous Exercise
After reading the solutions, students should test themselves on the same question types. The card below opens a set of solved practice questions with step-by-step answers for the Miscellaneous Exercise.
Practice Card: Solved Practice Questions for Class 11 Maths Sets Miscellaneous Exercise — attempt each question, then check the worked solution.
Other Exercises of Chapter 1 Sets
Chapter 1 Sets has five exercises before this Miscellaneous Exercise. Use the table to revise any earlier exercise.
| Exercise | NCERT Solutions link |
|---|---|
| Exercise 1.1 | Class 11 Maths Sets Exercise 1.1 Solutions |
| Exercise 1.2 | Class 11 Maths Sets Exercise 1.2 Solutions |
| Exercise 1.3 | Class 11 Maths Sets Exercise 1.3 Solutions |
| Exercise 1.4 | Class 11 Maths Sets Exercise 1.4 Solutions |
| Exercise 1.5 | Class 11 Maths Sets Exercise 1.5 Solutions |
More Class 11 Maths Sets Resources
Pair the solutions with the notes and the NCERT book PDF for full chapter revision. All three resources for Chapter 1 Sets are linked below.
| Resource | Link |
|---|---|
| Revision Notes | Class 11 Maths Sets Notes |
| Handwritten Notes | Class 11 Maths Sets Handwritten Notes |
| NCERT Book PDF | Class 11 Maths Sets NCERT Book PDF |
NCERT Solutions for Other Class 11 Maths Chapters
Students can move to the first exercise of any other Class 11 Maths chapter using the cross-sell table below.
| Chapter | NCERT Solutions |
|---|---|
| Chapter 1: Sets | You are here |
| Chapter 2: Relations and Functions | Relations and Functions Solutions |
| Chapter 3: Trigonometric Functions | Trigonometric Functions Solutions |
| Chapter 4: Complex Numbers and Quadratic Equations | Complex Numbers Solutions |
| Chapter 5: Linear Inequalities | Linear Inequalities Solutions |
| Chapter 6: Permutations and Combinations | Permutations and Combinations Solutions |
| Chapter 7: Binomial Theorem | Binomial Theorem Solutions |
| Chapter 8: Sequences and Series | Sequences and Series Solutions |
| Chapter 9: Straight Lines | Straight Lines Solutions |
| Chapter 10: Conic Sections | Conic Sections Solutions |
| Chapter 11: Introduction to Three Dimensional Geometry | Three Dimensional Geometry Solutions |
| Chapter 12: Limits and Derivatives | Limits and Derivatives Solutions |
| Chapter 13: Statistics | Statistics Solutions |
| Chapter 14: Probability | Probability Solutions |
Student Feedback
In a Collegedunia survey of 13,540 Class 11 students conducted before the 2026 exams, 71% of students said writing full proofs in the Miscellaneous Exercise was the hardest task in Chapter 1, while the subset questions were rated the most familiar.
FAQs on NCERT Solutions for Class 11 Maths Chapter 1 Sets Miscellaneous Exercise
Sets NCERT Solutions - Frequently Asked Questions
Ques. How many questions are there in Class 11 Maths Chapter 1 Sets Miscellaneous Exercise?
Ans. The Miscellaneous Exercise has 10 questions. They mix subset proofs, set identities, union, intersection, difference and counter-examples. Most questions ask for a short proof, so this exercise tests understanding of the whole Sets chapter.
Ques. How do you prove that two sets are equal?
Ans. To prove two sets A and B are equal, show that A ⊂ B and B ⊂ A. Take a general element of A and prove it lies in B, then do the reverse. This two-way method is used in several questions of the Miscellaneous Exercise.
Ques. Does A intersection B equal A intersection C mean B equals C?
Ans. No. A ∩ B = A ∩ C does not force B = C. Question 8 of the Miscellaneous Exercise asks for a counter-example, such as A = {1}, B = {2} and C = {3}, where both intersections are empty but B and C differ. One counter-example is enough to disprove the claim.
Ques. What set identities are used in the Sets Miscellaneous Exercise?
Ans. The Miscellaneous Exercise uses the absorption laws A ∪ (A ∩ B) = A and A ∩ (A ∪ B) = A, the identity A = (A ∩ B) ∪ (A − B), and De Morgan's laws. Students prove these by taking a general element and checking both sides.
Ques. Are the NCERT Solutions for Class 11 Maths Chapter 1 Sets Miscellaneous Exercise free to download?
Ans. Yes. The NCERT Solutions for Class 11 Maths Chapter 1 Sets Miscellaneous Exercise are free to download as a PDF from this page. Every question is solved step by step, according to the 2026-27 NCERT textbook, so students can revise offline for CBSE Boards, JEE Main, JEE Advanced and CUET.
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