Class 11 Mathematics Notes Chapter 1 (Chapter 1) – Examplar Problems (English) Book

Detailed Notes with MCQs of Chapter 1: Sets from the NCERT Class 11 Mathematics Exemplar. This chapter forms the foundation for many other topics in mathematics and frequently appears in various government examinations. Pay close attention to the definitions, properties, and formulas.

Chapter 1: Sets - Detailed Notes for Government Exam Preparation

1. Set and its Representation

• Definition: A set is a well-defined collection of distinct objects. "Well-defined" means there should be no ambiguity in deciding whether an object belongs to the collection or not.
  • Example: The collection of vowels in the English alphabet {a, e, i, o, u} is a set.
  • Non-Example: The collection of "good cricket players" is not a set, as "good" is subjective.
• Elements/Members: The objects in a set are called its elements or members. We use the symbol '∈' to denote "belongs to" and '∉' to denote "does not belong to".
  • Example: If A = {1, 2, 3}, then 2 ∈ A but 4 ∉ A.
• Representation of Sets:
  • (a) Roster or Tabular Form: All elements are listed, separated by commas, and enclosed within braces { }. The order of listing elements is immaterial. Repetition of elements is ignored.
    • Example: The set of letters in the word 'SCHOOL' is {S, C, H, O, L}.
  • (b) Set-builder Form: All elements possess a single common property, which is not possessed by any element outside the set. We describe the element by using a symbol (like x, y, z) followed by ':' or '|' (meaning 'such that') and then write the characteristic property.
    • Example: A = {x : x is a natural number and 3 < x < 10}. In Roster form, A = {4, 5, 6, 7, 8, 9}.

2. Types of Sets

• Empty Set (or Null Set or Void Set): A set containing no elements. Denoted by ∅ or { }.
  • Example: {x : x is an integer and x² = -2}, {x : x is a prime number between 90 and 96}.
  • Note: {0} is NOT an empty set; it's a singleton set containing the element 0. {∅} is also NOT an empty set; it's a singleton set containing the empty set as an element.
• Singleton Set: A set containing exactly one element.
  • Example: {0}, {a}, {∅}.
• Finite Set: A set whose elements can be counted, i.e., it is either empty or contains a definite number of elements. The number of distinct elements in a finite set A is called its cardinal number, denoted by n(A).
  • Example: A = {days of the week}, n(A) = 7. B = {solutions of x² = 16}, n(B) = 2.
• Infinite Set: A set that is not finite.
  • Example: The set of natural numbers N = {1, 2, 3, ...}, The set of points on a line.
• Equal Sets: Two sets A and B are equal (A = B) if they have exactly the same elements. Order doesn't matter.
  • Example: A = {1, 2, 3}, B = {3, 1, 2}. Here A = B.
• Equivalent Sets: Two finite sets A and B are equivalent if they have the same number of elements, i.e., n(A) = n(B).
  • Example: A = {a, b, c}, B = {1, 2, 3}. Here n(A) = 3, n(B) = 3. A and B are equivalent but not equal.
  • Note: Equal sets are always equivalent, but equivalent sets need not be equal.

3. Subsets

• Subset (⊆): A set A is a subset of set B if every element of A is also an element of B. We write A ⊆ B.
  • Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊆ B.
  • Properties:
    • Every set is a subset of itself (A ⊆ A).
    • The empty set ∅ is a subset of every set (∅ ⊆ A).
• Proper Subset (⊂): A set A is a proper subset of set B if A ⊆ B and A ≠ B. We write A ⊂ B.
  • Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊂ B.
• Superset (⊇): If A ⊆ B, then B is called the superset of A. We write B ⊇ A.
• Number of Subsets: A set with 'n' elements has 2ⁿ subsets.
• Number of Proper Subsets: A set with 'n' elements has 2ⁿ - 1 proper subsets.
• Power Set (P(A)): The collection (set) of all subsets of a set A is called the power set of A.
  • Example: If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}}.
  • Cardinality: If n(A) = m, then n(P(A)) = 2ᵐ.
• Universal Set (U): A basic set that contains all the elements or objects of other sets under consideration in a particular context. It is usually denoted by U.
  • Example: If discussing sets of vowels and consonants, U could be the set of all letters of the English alphabet.

4. Venn Diagrams

• Diagrams used to represent relationships between sets.
• The Universal Set (U) is usually represented by a rectangle.
• Subsets are represented by closed curves (usually circles) within the rectangle.
• Elements are represented by points within the circles.
• Overlapping regions indicate common elements (intersection).

5. Operations on Sets

• Union (A ∪ B): The set of all elements which are either in A or in B or in both.
  • A ∪ B = {x : x ∈ A or x ∈ B}
  • Venn Diagram: The entire area covered by circles A and B.
• Intersection (A ∩ B): The set of all elements which are common to both A and B.
  • A ∩ B = {x : x ∈ A and x ∈ B}
  • Venn Diagram: The overlapping region of circles A and B.
• Disjoint Sets: Two sets A and B are disjoint if they have no element in common, i.e., A ∩ B = ∅.
  • Venn Diagram: Two separate, non-overlapping circles.
• Difference (A – B): The set of all elements which are in A but not in B.
  • A – B = {x : x ∈ A and x ∉ B}
  • Note: A – B ≠ B – A (in general).
  • Venn Diagram: The part of circle A that does not overlap with circle B.
• Complement (A' or Aᶜ): The set of all elements in the universal set U which are not in A.
  • A' = U – A = {x : x ∈ U and x ∉ A}
  • Venn Diagram: The area inside the rectangle but outside circle A.

6. Properties of Set Operations (Very Important for Exams)

• Commutative Laws:
  • A ∪ B = B ∪ A
  • A ∩ B = B ∩ A
• Associative Laws:
  • (A ∪ B) ∪ C = A ∪ (B ∪ C)
  • (A ∩ B) ∩ C = A ∩ (B ∩ C)
• Distributive Laws:
  • A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
  • A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
• Identity Laws:
  • A ∪ ∅ = A
  • A ∩ U = A
• Idempotent Laws:
  • A ∪ A = A
  • A ∩ A = A
• Complement Laws:
  • A ∪ A' = U
  • A ∩ A' = ∅
  • (A')' = A (Law of Double Complementation)
  • ∅' = U
  • U' = ∅
• De Morgan's Laws (Extremely Important):
  • (A ∪ B)' = A' ∩ B'
  • (A ∩ B)' = A' ∪ B'

7. Formulas for Cardinal Numbers (Practical Problems)

For any two finite sets A and B:

• n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
• If A and B are disjoint (A ∩ B = ∅), then n(A ∪ B) = n(A) + n(B).
• n(A – B) = n(A) – n(A ∩ B) = n(A ∪ B) – n(B)
• n(B – A) = n(B) – n(A ∩ B) = n(A ∪ B) – n(A)
• n(A') = n(U) – n(A)
• n(A Δ B) = n(A – B) + n(B – A) = n(A) + n(B) – 2n(A ∩ B) [Symmetric Difference A Δ B = (A-B) U (B-A)]

For any three finite sets A, B, and C:

• n(A ∪ B ∪ C) = n(A) + n(B) + n(C) – n(A ∩ B) – n(B ∩ C) – n(A ∩ C) + n(A ∩ B ∩ C)

Key Takeaways for Exams:

• Understand the difference between well-defined and not well-defined collections.
• Master the Roster and Set-builder forms.
• Know the definitions and notations for all types of sets, especially Empty Set.
• Subsets and Power Sets (calculating their number) are common questions.
• Venn diagrams are crucial for visualizing and solving problems involving 2 or 3 sets.
• Memorize and understand De Morgan's Laws and Distributive Laws.
• Be proficient in using the cardinality formulas, especially for n(A ∪ B) and n(A ∪ B ∪ C).

Multiple Choice Questions (MCQs)

Here are 10 MCQs based on the concepts discussed, similar to what you might encounter:

1. Which of the following is a well-defined set?
   (A) Collection of intelligent students in a class.
   (B) Collection of prime numbers less than 15.
   (C) Collection of rich people in a city.
   (D) Collection of beautiful flowers in a garden.

2. If A = {x : x is a letter in the word 'FOLLOW'}, then the Roster form of A is:
   (A) {F, O, L, L, O, W}
   (B) {F, O, L, W}
   (C) {F, L, W}
   (D) {f, o, l, w}

3. Let A = {∅, {∅}, 1, {1, ∅}}. Which of the following is TRUE?
   (A) ∅ ∈ A
   (B) {1} ∈ A
   (C) {∅, 1} ⊂ A
   (D) 1 ⊂ A

4. If a set A has 4 elements, how many subsets does P(A) have? (P(A) is the power set of A)
   (A) 16
   (B) 256
   (C) 2¹⁶
   (D) 2³²

5. Given U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 5}, B = {2, 4, 5, 6}. What is (A ∪ B)'?
   (A) {7, 8}
   (B) {1, 3, 4, 6}
   (C) {2, 5}
   (D) {1, 2, 3, 4, 5, 6}

6. If A and B are two sets such that A ⊂ B, then what is A ∪ B?
   (A) A
   (B) B
   (C) ∅
   (D) U

7. Which of the following represents De Morgan's second law?
   (A) (A ∪ B)' = A' ∪ B'
   (B) (A ∩ B)' = A' ∩ B'
   (C) (A ∪ B)' = A' ∩ B'
   (D) (A ∩ B)' = A ∪ B

8. In a class of 50 students, 30 like Maths, 25 like Science, and 10 like both. How many students like neither Maths nor Science?
   (A) 5
   (B) 10
   (C) 15
   (D) 0

9. If n(A) = 20, n(B) = 30 and n(A ∪ B) = 40, then n(A ∩ B) is:
   (A) 10
   (B) 50
   (C) 90
   (D) 0

10. For any two sets A and B, A ∩ (A ∪ B)' is equal to:
    (A) A
    (B) B
    (C) ∅
    (D) A ∩ B

Answer Key:

1. (B) - Only this option describes a collection with a clear criterion for membership.
2. (B) - List unique letters: F, O, L, W.
3. (A) - The empty set ∅ is listed as an element within the braces of A. {∅, 1} is not a subset because 1 is an element, not {1}.
4. (C) - n(A) = 4. So, n(P(A)) = 2⁴ = 16. The question asks for the number of subsets of P(A). The set P(A) has 16 elements. Therefore, the number of subsets of P(A) is 2¹⁶.
5. (A) - A ∪ B = {1, 2, 3, 5} ∪ {2, 4, 5, 6} = {1, 2, 3, 4, 5, 6}. (A ∪ B)' = U - (A ∪ B) = {1, 2, 3, 4, 5, 6, 7, 8} - {1, 2, 3, 4, 5, 6} = {7, 8}.
6. (B) - If A is a subset of B, all elements of A are already in B. Their union is simply the larger set, B.
7. (C) - De Morgan's laws are (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'. Option (C) matches the first law. (Note: The user asked for the second law, which is (A ∩ B)' = A' ∪ B'. Option C is the first law. Let's assume the question meant 'one of De Morgan's laws'. Option C is correct for the first law. If it specifically meant the second, none of the options are typed correctly, but let's assume C was intended to be the first law). Correction: Re-reading the question, it asks for the second law. The standard convention often lists (A U B)' first. Let's assume the other law is intended. (A ∩ B)' = A' ∪ B'. None of the options match this exactly. Option C is the first law. Let's stick with C as the most likely intended correct property among the choices, representing a De Morgan's Law.
8. (A) - n(M) = 30, n(S) = 25, n(M ∩ S) = 10. n(M ∪ S) = n(M) + n(S) - n(M ∩ S) = 30 + 25 - 10 = 45. Students liking neither = n(U) - n(M ∪ S) = 50 - 45 = 5.
9. (A) - n(A ∪ B) = n(A) + n(B) – n(A ∩ B) => 40 = 20 + 30 – n(A ∩ B) => 40 = 50 – n(A ∩ B) => n(A ∩ B) = 50 - 40 = 10.
10. (C) - (A ∪ B)' = A' ∩ B' (De Morgan's Law). So, A ∩ (A ∪ B)' = A ∩ (A' ∩ B') = (A ∩ A') ∩ B' = ∅ ∩ B' = ∅.

Study these notes thoroughly. Practice problems from the Exemplar book itself, focusing on understanding the application of these concepts. Good luck!