📐 CHAPTER 1: SETS
Subject: Mathematics | Class: XI | Kendriya Vidyalaya Sangathan
📖 INTRODUCTION
A set is a well-defined collection of distinct objects. The objects in a set are called its elements or members.
🔹 Examples of Sets:
- Set of all natural numbers: N = {1, 2, 3, 4, ...}
- Set of all integers: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Set of all vowels in English alphabet: V = {a, e, i, o, u}
🔹 Representation of Sets
Sets can be represented in two ways:
- Roster Form (Tabular Form): Listing all elements inside curly braces.
Example: A = {2, 4, 6, 8} - Set-Builder Form: Describing the properties of elements.
Example: A = {x : x is an even natural number less than 10}
🔰 TYPES OF SETS
| Set Type | Definition | Example |
|---|---|---|
| Empty (Null) Set | Set with no elements | φ = { } or A = {x : x < 1 and x > 2} |
| Singleton Set | Set with exactly one element | A = {5} |
| Finite Set | Set with countable number of elements | A = {1, 2, 3} |
| Infinite Set | Set with uncountable number of elements | N = {1, 2, 3, ...} |
| Equal Sets | Sets having exactly the same elements | A = {1, 2, 3}, B = {3, 1, 2} |
| Equivalent Sets | Sets having same number of elements | A = {a, b, c}, B = {1, 2, 3} |
🔽 SUBSETS
A set A is a subset of set B if every element of A is also an element of B. We write A ⊂ B.
📌 Important Points:
- Every set is a subset of itself: A ⊂ A
- Empty set is a subset of every set: φ ⊂ A
- If A ⊂ B and B ⊂ A, then A = B
- Total number of subsets of a set with n elements = 2n
- Total number of proper subsets = 2n - 1
⚡ OPERATIONS ON SETS
1. Union of Sets (A ∪ B)
The set of all elements which belong to A or B or both.
2. Intersection of Sets (A ∩ B)
The set of all elements which belong to both A and B.
3. Difference of Sets (A - B)
The set of all elements which belong to A but not to B.
4. Complement of a Set (A′ or Ac)
The set of all elements of the universal set U which do not belong to A.
📊 VENN DIAGRAMS
Venn diagrams are pictorial representations of sets using closed figures (usually circles).
| Operation | Venn Diagram Representation |
|---|---|
| A ∪ B | Shaded region = all areas covered by circles A and B |
| A ∩ B | Shaded region = overlapping area of circles A and B |
| A - B | Shaded region = part of A outside B |
| A′ | Shaded region = area outside circle A (within U) |
📜 IMPORTANT LAWS OF SETS
🔹 Idempotent Laws
- A ∪ A = A
- A ∩ A = A
🔹 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)
🔹 De Morgan's Laws
- (A ∪ B)′ = A′ ∩ B′
- (A ∩ B)′ = A′ ∪ B′
📝 1 MARK QUESTIONS
Q76. Write the set {5, 25, 125, 625} in set-builder form.
Answer: {x : x = 5n, n ∈ N, 1 ≤ n ≤ 4}
Q77. If n(A) = 2, then find n[P(P(A))].
Answer: n(A) = 2 ⇒ n(P(A)) = 22 = 4
n[P(P(A))] = 24 = 16
Q78. Define equal sets with example.
Answer: Two sets are equal if they have exactly the same elements.
Example: A = {1, 2, 3} and B = {3, 1, 2} are equal sets.
Q79. Write {x : x ∈ R, -4 < x ≤ 6} as interval.
Answer: (-4, 6]
Q80. If A = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6, 8}, find A - B.
Answer: A - B = {1, 3, 5}
Q81. Write the set A = {x : x ∈ R, 2x + 11 = 15} in roster form.
Answer: 2x + 11 = 15 ⇒ 2x = 4 ⇒ x = 2
∴ A = {2}
Q82. Define disjoint sets with example.
Answer: Two sets are disjoint if they have no common elements.
Example: A = {1, 2, 3} and B = {4, 5, 6} are disjoint sets.
Q83. What universal set would you propose for the set of rectangles?
Answer: The set of all quadrilaterals.
Q84. If A and B are two sets such that A ⊂ B, then what is A ∪ B?
Answer: A ∪ B = B
Q85. Write down the power set for the set {a, b}.
Answer: P({a, b}) = {φ, {a}, {b}, {a, b}}
Q86. If S = {x : x is a positive multiple of 3 less than 100} and P = {x : x is a prime number less than 20}, then find n(S) + n(P).
Answer: S = {3, 6, 9, ..., 99} ⇒ n(S) = 33
P = {2, 3, 5, 7, 11, 13, 17, 19} ⇒ n(P) = 8
n(S) + n(P) = 33 + 8 = 41
Q87. Two finite sets have m and n elements. The total number of subsets of the first set is 56 more than the total number of subsets of the second set. Find the values of m and n.
Answer: 2m - 2n = 56
2n(2m-n - 1) = 56
56 = 8 × 7 = 23 × 7
∴ n = 3 and 2m-n - 1 = 7 ⇒ 2m-n = 8 = 23
∴ m - n = 3 ⇒ m = 6
m = 6, n = 3
Q88. Let S = set of all triangles, P = set of all isosceles triangles, Q = set of all equilateral triangles, R = set of all right-angled triangles. What do the sets P ∩ Q and R - P represent respectively?
Answer: P ∩ Q = Set of all equilateral triangles (since all equilateral triangles are isosceles)
R - P = Set of all right-angled triangles which are not isosceles
Q89. Let V = {a, e, i, o, u}, V - B = {e, o} and B - V = {k}. Find the set B.
Answer: V - B = {e, o} ⇒ e, o ∈ V and e, o ∉ B
B - V = {k} ⇒ k ∈ B and k ∉ V
Also, elements of V not in (V - B) are in B
∴ B = {a, i, u, k}
Q90. If A = {a, {b}}, then find P(A).
Answer: A has 2 elements: a and {b}
P(A) = {φ, {a}, {{b}}, {a, {b}}}
📝 2 MARKS QUESTIONS
Q91. Write down all the subsets of the set {1, 2, 3}.
Answer: φ, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
Q92. Draw appropriate Venn diagram for each of the following:
(a) (A ∪ B)′ (b) A - B
Answer:
(a) (A ∪ B)′ = Area outside both A and B
(b) A - B = Area of A not overlapping with B
Q93. Write the set A = {5, 25, 125, 625} in set-builder form.
Answer: A = {x : x = 5n, n ∈ N, 1 ≤ n ≤ 4}
Q94. Prove that A = B if P(A) = P(B).
Answer: If P(A) = P(B), then every subset of A is a subset of B and vice versa.
Since A ⊂ P(A) and B ⊂ P(B), we get A ⊂ B and B ⊂ A
∴ A = B
Q95. A survey shows that 63% of Indians like cheese whereas 76% like apples. If x% of Indians like cheese and apples, find the value of x.
Answer: Let total = 100%
n(C) = 63, n(A) = 76
n(C ∪ A) ≤ 100
n(C) + n(A) - n(C ∩ A) ≤ 100
63 + 76 - x ≤ 100
139 - x ≤ 100 ⇒ x ≥ 39
Also, x ≤ min(63, 76) = 63
∴ 39 ≤ x ≤ 63
Q96. If n(A) = 4, how many proper subsets does the set A have?
Answer: Total subsets = 24 = 16
Proper subsets = 16 - 1 = 15
Q97. Using properties of sets prove that (A ∪ B)′ = A′ ∩ B′.
Answer: By De Morgan's Law:
(A ∪ B)′ = {x : x ∉ (A ∪ B)}
= {x : x ∉ A and x ∉ B}
= {x : x ∈ A′} ∩ {x : x ∈ B′}
= A′ ∩ B′
Q98. Using Venn Diagram prove that: A ∩ (B - C) = (A ∩ B) - (A ∩ C).
Answer: A ∩ (B - C) = A ∩ (B ∩ C′)
= (A ∩ B) ∩ C′
= (A ∩ B) - C
= (A ∩ B) - (A ∩ C) [since (A ∩ B) ∩ (A ∩ C) = A ∩ B ∩ C]
Q99. For all sets A, B and C, if A ⊂ B, then A ∪ C ⊂ B ∪ C.
Answer: Let x ∈ A ∪ C ⇒ x ∈ A or x ∈ C
If x ∈ A, then x ∈ B (since A ⊂ B)
If x ∈ C, then x ∈ C
Thus in both cases, x ∈ B ∪ C
∴ A ∪ C ⊂ B ∪ C
Q100. For all sets A and B, (A - B) ∪ (A ∩ B) = A.
Answer: (A - B) ∪ (A ∩ B)
= (A ∩ B′) ∪ (A ∩ B)
= A ∩ (B′ ∪ B)
= A ∩ U
= A
📝 4 MARKS QUESTIONS
Q101. If A = {3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, C = {11, 13, 15}, D = {15, 17}, find:
(a) (A ∪ B) ∩ C (b) C - D (c) (A ∩ B) ∩ (B ∪ C) (d) B ∩ D
Answer:
(a) A ∪ B = {3, 5, 7, 9, 11, 13}
(A ∪ B) ∩ C = {11, 13}
(b) C - D = {11, 13} (since 15 is common)
(c) A ∩ B = {7, 9, 11}
B ∪ C = {7, 9, 11, 13, 15}
(A ∩ B) ∩ (B ∪ C) = {7, 9, 11}
(d) B ∩ D = φ (no common elements)
Q102. If S and T are two sets such that S has 21 elements, T has 32 elements, and S ∩ T has 11 elements, how many elements does S ∪ T have?
Answer: n(S ∪ T) = n(S) + n(T) - n(S ∩ T)
= 21 + 32 - 11
= 42
Q103. In a survey of 100 students, the number of students studying various languages were found to be: English only 18, English but not Hindi 23, English and Sanskrit 8, English 26, Sanskrit 48, Sanskrit and Hindi 8, no language 24. Find:
(i) How many students were studying Hindi?
(ii) How many students were studying English and Hindi?
(iii) How many students were studying Sanskrit only?
Answer: Let E = English, H = Hindi, S = Sanskrit
Total = 100, n(E) = 26, n(S) = 48, n(E ∩ S) = 8
English only = 18
(iii) Sanskrit only = n(S) - n(E ∩ S) - n(H ∩ S) + n(E ∩ H ∩ S)
= 48 - 8 - 8 + n(E ∩ H ∩ S)
= 32 + n(E ∩ H ∩ S)
(i) Hindi = n(H) = ?
(ii) English and Hindi = n(E ∩ H) = ?
Using the given data, solving we get:
n(H) = 14
n(E ∩ H) = 3
n(Sanskrit only) = 34
Q104. Define the following with the help of an example:
(a) Null set (b) Singleton Set (c) Sub-Set (d) Power Set
Answer:
(a) Null set: A set with no elements. Example: {x : x is a natural number less than 1} = φ
(b) Singleton set: A set with exactly one element. Example: {5}
(c) Sub-set: A set A is a subset of B if every element of A is in B. Example: {1, 2} ⊂ {1, 2, 3}
(d) Power set: The set of all subsets of a set. Example: P({a, b}) = {φ, {a}, {b}, {a, b}}
📝 TRUE/FALSE QUESTIONS
Q105. In each of the following, determine whether the given statement is true or false. If true prove it, and if false, give an example.
(a) If x ∈ A and A ∈ B, then x ∈ B
(b) If x ∈ A and A ⊂ B, then x ∈ B
Answer:
(a) False. Example: A = {1, 2}, B = {A, 3}. Then x = 1 ∈ A and A ∈ B, but x ∉ B.
(b) True. Since A ⊂ B, every element of A is also in B. Thus x ∈ A ⇒ x ∈ B.
Q106. In a survey of 25 students of a school it was found that 15 study Mathematics, 12 study Physics, 11 study Chemistry, 9 study both Mathematics & Physics, 4 study both Physics & Chemistry, 5 study both Chemistry and Mathematics and 3 students all of the above three subjects. Find the number of students who study:
(a) Only Mathematics
(b) At least one of the three subjects
Answer: Let M = Mathematics, P = Physics, C = Chemistry
n(M) = 15, n(P) = 12, n(C) = 11
n(M ∩ P) = 9, n(P ∩ C) = 4, n(C ∩ M) = 5, n(M ∩ P ∩ C) = 3
(a) Only Mathematics = n(M) - n(M ∩ P) - n(M ∩ C) + n(M ∩ P ∩ C)
= 15 - 9 - 5 + 3 = 4
(b) At least one = n(M ∪ P ∪ C)
= n(M) + n(P) + n(C) - n(M ∩ P) - n(P ∩ C) - n(C ∩ M) + n(M ∩ P ∩ C)
= 15 + 12 + 11 - 9 - 4 - 5 + 3 = 23
Q107. In a group of students, 100 students know Hindi, 50 know English and 25 know both. Each of the students know either Hindi or English. How many students are there in the group?
Answer: n(H ∪ E) = n(H) + n(E) - n(H ∩ E)
= 100 + 50 - 25 = 125
Q108. If n(A - B) = 18, n(A ∪ B) = 70 and n(A ∩ B) = 25, then find n(B).
Answer: n(A - B) = n(A) - n(A ∩ B)
⇒ n(A) = 18 + 25 = 43
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
70 = 43 + n(B) - 25
n(B) = 70 - 43 + 25 = 52
Q109. If L = {1, 2, 3, 4}, M = {3, 4, 5, 6} and N = {1, 3, 5}, then verify that L - (M ∪ N) = (L - M) ∩ (L - N).
Answer: M ∪ N = {3, 4, 5, 6} ∪ {1, 3, 5} = {1, 3, 4, 5, 6}
L - (M ∪ N) = {1, 2, 3, 4} - {1, 3, 4, 5, 6} = {2}
L - M = {1, 2, 3, 4} - {3, 4, 5, 6} = {1, 2}
L - N = {1, 2, 3, 4} - {1, 3, 5} = {2, 4}
(L - M) ∩ (L - N) = {1, 2} ∩ {2, 4} = {2}
∴ L - (M ∪ N) = (L - M) ∩ (L - N) ✓
Q110. If A, B and C be sets. Then, show that A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Answer: This is the Distributive Law of Intersection over Union.
Let x ∈ A ∩ (B ∪ C)
⇒ x ∈ A and (x ∈ B or x ∈ C)
⇒ (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C)
⇒ x ∈ (A ∩ B) or x ∈ (A ∩ C)
⇒ x ∈ (A ∩ B) ∪ (A ∩ C)
Conversely, let x ∈ (A ∩ B) ∪ (A ∩ C)
⇒ x ∈ (A ∩ B) or x ∈ (A ∩ C)
⇒ (x ∈ A and x ∈ B) or (x ∈ A and x ∈ C)
⇒ x ∈ A and (x ∈ B or x ∈ C)
⇒ x ∈ A ∩ (B ∪ C)
∴ A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) ✓
🧠 ASSERTION & REASON TYPE QUESTIONS
Directions: Each of these questions contains two statements, Assertion and Reason. Each of these questions also has four alternative choices, only one of which is the correct answer.
Q56. Assertion: The number of non-empty subsets of the set {a, b, c, d} are 15.
Reason: Number of non-empty subsets of a set having n elements are 2n - 1.
Answer: (a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
n = 4, non-empty subsets = 24 - 1 = 15 ✓
Q57. Suppose A, B and C are three arbitrary sets and U is a universal set.
Assertion: If B = U - A, then n(B) = n(U) - n(A).
Reason: If C = A - B, then n(C) = n(A) - n(B).
Answer: (b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
Reason is true only when B ⊂ A.
Q58. Assertion: Let A = {1, {2, 3}}, then P(A) = {{1}, {2, 3}, φ, {1, {2, 3}}}.
Reason: Power set is set of all subsets of A.
Answer: (a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
A has 2 elements, so P(A) has 22 = 4 elements ✓
Q59. Assertion: The subsets of the set {1, {2}} are {}, {1}, {{2}} and {1, {2}}.
Reason: The total number of proper subsets of a set containing n elements is 2n - 1.
Answer: (b) Assertion is correct, reason is correct; reason is not a correct explanation for assertion.
Reason gives total proper subsets, not the list of subsets.
Q60. Assertion: For any two sets A and B, A - B ⊂ B′
Reason: If A be any set, then A ∩ A′ = φ
Answer: (a) Assertion is correct, reason is correct; reason is a correct explanation for assertion.
x ∈ A - B ⇒ x ∈ A and x ∉ B ⇒ x ∈ B′ ✓
📌 KEY CONCEPTS SUMMARY
| Concept | Formula / Definition |
|---|---|
| Number of subsets | 2n |
| Number of proper subsets | 2n - 1 |
| Union of sets | A ∪ B = {x : x ∈ A or x ∈ B} |
| Intersection of sets | A ∩ B = {x : x ∈ A and x ∈ B} |
| Difference of sets | A - B = {x : x ∈ A and x ∉ B} |
| Complement of a set | A′ = U - A |
| Symmetric difference | A Δ B = (A - B) ∪ (B - A) |
| De Morgan's Law | (A ∪ B)′ = A′ ∩ B′ |
| De Morgan's Law | (A ∩ B)′ = A′ ∪ B′ |
📚 Kendriya Vidyalaya Sangathan | Class XI Mathematics
Chapter 1: Sets | Student Support Material