Free Study Material Sets – Types of Sets

Sets – Class 11 Mathematics

Chapter 1: Sets (Class 11 CBSE/ISC)

Comprehensive Study Material

Definition

A set is a well-defined collection of distinct objects, considered as an object in its own right.

Definition & Representation of Sets

Well-defined: The objects in a set must be clearly defined so that we can determine whether a given object belongs to the set or not.

  • Example: “The collection of tall students in a class” is not a set because “tall” is subjective.
  • Example: “The collection of students whose height is more than 170 cm” is a set.

Representation of Sets: There are two common ways to represent a set:

  1. Roster Form (Tabular Form): All elements are listed within curly braces { }, separated by commas.
    • Example: A = {1, 2, 3, 4, 5}
    • Note: Order does not matter; repetitions are not counted.
  2. Set-Builder Form: A property that all elements satisfy is described.
    • Example: A = {x : x is a natural number and 1 ≤ x ≤ 5} or A = {x ∈ ℕ : 1 ≤ x ≤ 5}.
Real-Life Relevance

Sets are used in database searches (e.g., filtering products by multiple attributes), in genetics (set of genes), and in everyday grouping of items.

Roster Form {1,2,3} Set-Builder {x: 1<x<4, x∈ℕ}
converts to / equivalent representations

Types of Sets: Empty, Finite, Infinite

  • Empty Set (Null/Void Set): A set with no elements. Denoted by ∅ or {}.
    • Example: A = {x : x is a natural number less than 1} = ∅.
  • Finite Set: A set with a definite (countable) number of elements.
    • Example: B = {a, e, i, o, u} has 5 elements.
  • Infinite Set: A set with an unlimited number of elements.
    • Example: C = {1, 2, 3, …} (Natural numbers) is infinite.

Equal Sets & Subsets

Definition

Two sets A and B are equal if they have exactly the same elements. We write A = B.

  • Example: A = {1,2,3}, B = {3,2,1} ⇒ A = B.
Definition

A set A is a subset of set B if every element of A is also an element of B. Denoted A ⊆ B.

  • Proper Subset: If A ⊆ B and A ≠ B, then A is a proper subset of B, denoted A ⊂ B.
  • Important: ∅ is a subset of every set.
  • Important: Every set is a subset of itself.
B A A ⊂ B
Important Formulas / Facts
  • Number of subsets of a set with n elements = 2ⁿ.
  • Number of proper subsets = 2ⁿ – 1.
  • ∅ ⊆ A for any set A.
  • If A ⊆ B and B ⊆ A, then A = B.

Solved Examples

Example 1
Which of the following are sets? Justify your answer.
(i) The collection of all months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
Solution: (i) This is a set because we can list the months: January, June, July. It is well-defined.
(ii) This is not a set because “most talented” is subjective and not well-defined.
Example 2
Write the set {x: x is a positive integer and x² < 40} in roster form.
Solution: Positive integers whose square is less than 40: 1,2,3,4,5,6 → {1,2,3,4,5,6}.
Example 3
State whether A = B where A = {2,4,6,8} and B = {x: x is an even integer and 2 ≤ x ≤ 8}.
Solution: B in roster form is {2,4,6,8}. Since both have identical elements, A = B.

Additional Solved Examples

Extra Example 1
If A = {1,2,3}, list all subsets of A. Also find the number of proper subsets.
Solution: Subsets: ∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}.
Number of proper subsets = 2³−1 = 7.
Extra Example 2
Let A = {x: x is a prime number less than 10} and B = {2,3,5,7}. Is A ⊆ B? Is B ⊆ A?
Solution: A = {2,3,5,7} ⇒ A ⊆ B and B ⊆ A, therefore A = B.
Extra Example 3
Classify as finite or infinite:
(a) {x ∈ ℕ : x > 100}   (b) {x ∈ ℤ : x² = 4}
Solution: (a) Infinite   (b) Finite: {2, -2}
Extra Example 4
Prove that the empty set is a subset of any set A.
Solution: By definition, ∅ has no elements. The statement “every element of ∅ is in A” is vacuously true. Hence ∅ ⊆ A.
Extra Example 5
If A = {1, {2,3}, 4}, which are true? (i) 2 ∈ A (ii) {2,3} ∈ A (iii) {2,3} ⊆ A
Solution: (i) False (ii) True (iii) False

Practice Problems

  1. Write the set {x: x is an integer and -3 ≤ x < 4} in roster form.
  2. Identify whether “the collection of all even prime numbers” is a set or not. Justify.
  3. List all subsets of C = {a, b}. How many are proper?
  4. If D = {1,3,5} and E = {x: x is an odd natural number less than 6}, is D = E? Why?
  5. Classify as finite or infinite: {x ∈ ℝ : 1 < x < 2}.
  6. Let P = ∅. Is P ⊆ Q for any set Q? Explain.
  7. If a set has 4 elements, how many subsets does it have? How many non-empty?
  8. Give an example of an infinite set from your daily life.
  9. Write the set of all vowels in the English alphabet in set-builder form.
  10. If A = {1,2,3,4}, B = {2,4,6}, is A ⊆ B? Is B ⊆ A?
  11. Find the number of proper subsets of {x: x is a day of the week}.
  12. Determine if {0} is an empty set. Justify.
  13. Represent integers between -2 and +3 (inclusive) in roster and set-builder form.
  14. True or False: ∅ ∈ {∅, {∅}}.
  15. Give two examples of equal sets from your classroom.

Solutions to Practice Problems

  1. Solution: { -3, -2, -1, 0, 1, 2, 3 }
  2. Solution: The only even prime number is 2 → {2} is a set.
  3. Solution: Subsets: ∅, {a}, {b}, {a,b}. Proper subsets = 3.
  4. Solution: E = {1,3,5} ⇒ D = E.
  5. Solution: Infinite (uncountably many real numbers).
  6. Solution: Yes, ∅ is subset of any set (vacuously true).
  7. Solution: 2⁴ = 16 subsets; non-empty = 15.
  8. Solution: The set of all natural numbers, or all points on a line.
  9. Solution: {x : x is a vowel in English alphabet}.
  10. Solution: Neither A⊆B nor B⊆A.
  11. Solution: 2⁷ − 1 = 127.
  12. Solution: No, {0} contains element 0, not empty.
  13. Solution: Roster: {-2,-1,0,1,2,3}; Set-builder: {x∈ℤ : -2 ≤ x ≤ 3}
  14. Solution: True (∅ is an element of that set).
  15. Solution: e.g., set of students born in January = set of students with birthday in Jan.

MCQ Test

Instructions: Choose the correct option for each question.

  1. Which of the following is a set?
    A) all good movies   B) all prime numbers less than 20   C) tall students   D) honest people
  2. Roster form of {x: x∈ℕ, x²=9} is:
    A) {9}   B) {3}   C) {3,-3}   D) {3,9}
  3. The empty set is a subset of:
    A) only finite   B) only infinite   C) every set   D) no set
  4. How many subsets does a set with 3 elements have?
    A) 3   B) 6   C) 8   D) 9
  5. If A={1,2,3} and B={3,1,2}, then:
    A) A⊂B   B) B⊂A   C) A=B   D) none
  6. Which is an infinite set?
    A) {x∈ℕ: x≤100}   B) {x∈ℤ: x²=4}   C) {x∈ℝ: 0<x<1}   D) vowels
  7. Number of proper subsets of {1,2,3,4} is:
    A) 15   B) 16   C) 14   D) 12
  8. If A={1,{2},3}, then which is true?
    A) 2∈A   B) {2}∈A   C) {2}⊆A   D) 2⊆A
  9. The set {x: x∈ℕ, 1<x<2} is:
    A) Finite   B) Infinite   C) Empty set   D) Singleton
  10. If A⊆B and B⊆A, then:
    A) A⊂B   B) B⊂A   C) A=B   D) A∩B=∅

Answers to MCQs

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