Class 9 Maths Sets Notes

Grade 9 Mathematics — Advanced Level Notes

Chapter 1 — Sets

1. Meaning of a Set

A set is a clearly defined collection of objects. The objects are called elements or members.

The important idea is well-definedness: it must be possible to decide unambiguously whether an object belongs to the collection.

Example:

{2, 4, 6, 8} is a set.
“The most interesting books” is not well-defined because “interesting” can mean different things to different people.

2. Representation of Sets

Two important forms are:

Roster form: list the elements inside braces.

A = {1, 2, 3, 4}

Set-builder form: describe the common property of the elements.

A = {x : x is a natural number less than 5}

Symbols:

x ∈ A → x belongs to A
x ∉ A → x does not belong to A

The order in which elements are written does not matter, and repeated elements are counted only once.

3. Finite and Infinite Sets

A set is finite if it has a definite number of elements.

A set is infinite if its elements cannot be completely counted/listed.

For example, the set of students in a class is finite, whereas the set of natural numbers is infinite.

4. Special Types of Sets

Empty set: contains no element.

∅ or { }

Equal sets: two sets are equal when they contain exactly the same elements, irrespective of order or repetition.

5. Subsets

A ⊆ B means every element of A is also an element of B.

If A ⊆ B but A ≠ B, then A is a proper subset of B.

A key fact:

The empty set is a subset of every set.

6. Cardinality

The cardinality of a set is its number of distinct elements.

It is written as:

n(A)

If A = {2, 4, 6, 8}, then n(A) = 4.

7. Power Set

The power set P(A) is the set containing all subsets of A.

If a set contains n elements:

n[P(A)] = 2ⁿ

Therefore, the number of proper subsets is:

2ⁿ − 1

Remember that elements such as {a} and {a,b} can themselves be elements of a power set; do not confuse a set with an element of that set.

8. Universal Set and Venn Diagrams

The universal set U contains all objects under consideration.

A Venn diagram represents sets visually, generally using circles inside a rectangle representing U.

It is especially useful for understanding:

subsets
union
intersection
complement
differences
counting problems

9. Operations on Sets

Union

A ∪ B

contains elements belonging to A, B, or both.

Intersection

A ∩ B

contains elements common to both A and B.

Disjoint sets

If two sets have no common element:

A ∩ B = ∅

Difference

A − B

contains elements belonging to A but not B.

A − B = {x : x ∈ A and x ∉ B}

Complement

With respect to universal set U:

A′ = U − A

It contains elements of U that are outside A.

10. De Morgan’s Laws

Two important relationships are:

(A ∪ B)′ = A′ ∩ B′

(A ∩ B)′ = A′ ∪ B′

Venn diagrams provide a useful way to verify these identities.

11. Counting with Sets

For two finite sets:

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

The subtraction is necessary because common elements have initially been counted twice.

Other useful relations:

n(A − B) = n(A) − n(A ∩ B)

n(A′ ∩ B′) = n(U) − n(A ∪ B)

n[(A − B) ∪ (B − A)] = n(A) + n(B) − 2n(A ∩ B)

For three sets:

n(A ∪ B ∪ C)
= n(A) + n(B) + n(C)
− n(A ∩ B) − n(B ∩ C) − n(C ∩ A)
+ n(A ∩ B ∩ C)

Exam Focus

Be particularly comfortable with:

roster ↔ set-builder conversion
subset vs element
power sets
Venn diagrams
complement
De Morgan’s laws
two-set and three-set counting problems

Chapter 1 — Sets: Question Bank

A. Multiple Choice Questions (MCQs)

1. Which of the following represents a well-defined set?

A. The most beautiful flowers
B. The cleverest students
C. The prime numbers less than 20
D. The most interesting books

2. If A = {2, 4, 6, 8}, which statement is correct?

A. 3 ∈ A
B. 4 ∈ A
C. 6 ∉ A
D. 8 ∉ A

3. Which form represents a set by listing all its elements?

A. Set-builder form
B. Roster form
C. Cartesian form
D. Venn form

4. If A = {a, b, c}, the number of elements in A is:

A. 2
B. 3
C. 4
D. 6

5. Which set is empty?

A. {0}
B. {∅}
C. {x : x is a natural number less than 0}
D. {1}

6. If A ⊆ B, which statement must be true?

A. Every element of B belongs to A
B. Every element of A belongs to B
C. A and B have no common elements
D. A and B must be equal

7. If a set has 4 elements, its power set contains:

A. 4 subsets
B. 8 subsets
C. 12 subsets
D. 16 subsets

8. If A ∩ B = ∅, then A and B are:

A. Equal sets
B. Universal sets
C. Disjoint sets
D. Equivalent sets

9. The set containing all objects under consideration is called the:

A. Empty set
B. Universal set
C. Power set
D. Subset

10. Which operation contains elements common to both A and B?

A. A ∪ B
B. A − B
C. A ∩ B
D. A′

B. Fill in the Blanks

11. A set is a __________ defined collection of objects.

12. The objects belonging to a set are called its __________.

13. Sets are generally denoted by __________ letters.

14. The symbol ∈ means __________.

15. The number of elements in a finite set A is denoted by __________.

16. A set containing no elements is called the __________ set.

17. The set of all subsets of A is called the __________ of A.

18. If A has n elements, then its power set has __________ elements.

19. The symbol A ∩ B represents the __________ of A and B.

20. If A ∩ B = ∅, A and B are called __________ sets.

21. A − B contains elements that belong to __________ but not to B.

22. The complement of A with respect to U is written as __________.

C. True or False

23. The order of elements in a set affects the set.

24. Repeating an element in a roster changes the set.

25. Every empty set is a subset of every set.

26. If A ⊆ B and B ⊆ A, then A and B are equal.

27. Two disjoint sets have at least one common element.

28. The power set of a set with 3 elements has 6 elements.

29. The complement of A depends on the universal set being considered.

30. A ∩ B contains elements belonging to A or B.

D. Match the Following

Column A	Column B
31. `A ∪ B`	a. Elements common to A and B
32. `A ∩ B`	b. Elements of A not in B
33. `A − B`	c. All elements outside A within U
34. `A′`	d. Elements in A or B or both
35. `∅`	e. Set containing no elements

E. Identify the Type

36. A = {1, 3, 5, 7, 9}
Is A finite or infinite?

37. B = {x : x is a natural number}
Is B finite or infinite?

38. C = {2, 4, 6} and D = {6, 4, 2}
What is the relationship between C and D?

39. P = {1,2} and Q = {3,4}
What type of relationship do P and Q have with respect to intersection?

40. A = {1,2} and B = {1,2,3,4}
What is the relationship between A and B?

F. Short Answer Questions

41. Explain what makes a collection a well-defined set.

42. Write the set of vowels in the English alphabet in roster form.

43. Write the positive even integers less than 10 in set-builder form.

44. Explain the difference between an element and a subset.

45. Find the number of subsets and proper subsets of a set containing 5 elements.

46. If
A = {1,2,3,4} and B = {3,4,5,6},
find A ∪ B and A ∩ B.

47. If
A = {2,4,6,8} and B = {4,8},
determine whether B is a subset of A.

48. Explain the difference between A − B and B − A.

G. Venn Diagram / Application Questions

49. In a class, some students study Mathematics and some study Science. Explain how a Venn diagram can represent students who study both subjects.

50. If n(A) = 28, n(B) = 35, and n(A ∩ B) = 12, find n(A ∪ B).

51. In a group of 60 students, 35 like cricket, 28 like football, and 15 like both. How many like at least one of the two sports?

52. If the universal set contains 50 elements and a set A contains 31 elements, find n(A′).

53. For two finite sets A and B, explain why the number of elements in their union requires subtracting the number of common elements.

H. Higher-Order Thinking Questions

54. A set has exactly 32 subsets. How many elements does the original set contain?

55. Two sets have 7 and 5 elements respectively. Their intersection has 3 elements. Find the number of elements in their union.

56. If A ⊆ B and B ⊆ C, what relationship can you conclude between A and C? Explain.

57. Can two non-empty sets have an empty intersection? Give a suitable example and explain.

58. A student claims that {1,2,3} and {3,2,1,1} are different sets. Is the student correct? Justify.

Answer Key

MCQs

Fill in the Blanks

well
elements/members
capital
belongs to
n(A)
empty/null
power set
2ⁿ
intersection
disjoint
A
A′

True/False

False
False
True
True
False
False
True
False

Match

31–d
32–a
33–b
34–c
35–e

Selected Numerical Answers

32 subsets; 31 proper subsets
A ∪ B = {1,2,3,4,5,6}; A ∩ B = {3,4}
51
48
19
5 elements
9 elements