Exercise 7: Set
1.Which of the following is a subset of any set?
a) {0}
b) {Ø}
c) Ø
d) (Ø)
Solution: The empty set (Ø) is a subset of any set.
Correct answer: c) Ø
Answer the questions from 2 to 5 in respect of the adjoining Venn diagram:
Elements from the Venn diagram:
Universal set U = {1, 2, 3, 4, 5, 6, 7, 8}
Set A = {1, 2, 3, 4}
Set B = {2, 3, 5, 6}
Set C = {3, 4, 6, 7}
2. Which one is Universal set?
(a) A
(b) B
(c) A ∪ B
(d) U
Solution: In a Venn diagram, the rectangle represents the universal set, which is denoted by the symbol U.
Correct answer: (d) U
3. Which one is the set Bᶜ?
(a) {5, 6, 7, 8}
(b) {2, 3, 5, 6}
(c) {1, 4, 7, 8}
(d) {3, 6}
Solution: Bᶜ = U – B
= {1, 2, 3, 4, 5, 6, 7, 8} – {2, 3, 5, 6}
= {1, 4, 7, 8}
Correct answer: (c) {1, 4, 7, 8}
4. Which one is the set A ∩ B?
(a) {2, 3}
(b) {2, 3, 5, 6}
(c) {3, 4, 6, 7}
(d) {2, 3, 4, 5, 6, 7}
Solution: A ∩ B = {1, 2, 3, 4} ∩ {2, 3, 5, 6}
= {2, 3}
Correct answer: (a) {2, 3}
5. Which one is the set A ∪ B?
(a) {1, 2, 3, 4, 5, 6}
(b) {5, 6, 7}
(c) {8}
(d) {3}
Solution:
A ∪ B = {1, 2, 3, 4} ∪ {2, 3, 5, 6}
= {1, 2, 3, 4, 5, 6}
Correct answer: (a) {1, 2, 3, 4, 5, 6}
6. Express the following sets in tabular form:
(a) {x : x is odd number and 3 < x < 15}
Solution:
The odd numbers greater than 3 and less than 15 are: 5, 7, 9, 11, 13
Required set: {5, 7, 9, 11, 13}
(b) {x : x is a prime factor of 48}
Solution:
All factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
Among these, the prime numbers are: 2, 3
Required set: {2, 3}
(c) {x : x is a multiple of 3 and x < 36}
Solution:
The positive multiples of 3 less than 36 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33
Required set: {3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33}
(d) {x : x is an integer and x² < 10}
Solution:
Integers whose square is less than 10:
Square of 0 = 0 < 10
Square of (±1) = 1 < 10
Square of (±2) = 4 < 10
Square of (±3) = 9 < 10
Square of (±4) = 16 (which is not less than 10)
Required set: {-3, -2, -1, 0, 1, 2, 3}
7. Express the following sets in set-builder form.
(a) {3, 4, 5, 6, 7, 8}
Solution:
The elements of the set are natural numbers that are greater than 2 and less than 9.
Required set: {x : x is a natural number and 2 < x < 9}
(b) {4, 8, 12, 16, 20, 24}
Solution:
The elements are multiples of 4 and less than or equal to 24.
Required set: {x : x is a multiple of 4 and x ≤ 24}
(c) {7, 11, 13, 17}
Solution:
The elements are prime numbers that are greater than 5 and less than 19.
Required set: {x : x is a prime number and 5 < x < 19}
8. If A = {1, 2, 3}, B = {2, a}, and C = {a, b}, find the following sets:
(a) A ∪ B
Solution: A ∪ B = {1, 2, 3} ∪ {2, a}
= {1, 2, 3, a}
(b) B ∩ C
Solution: B ∩ C = {2, a} ∩ {a, b}
= {a}
(c) A ∩ (B ∪ C)
Solution: B ∪ C = {2, a} ∪ {a, b} = {2, a, b}
Therefore,
A ∩ (B ∪ C) = {1, 2, 3} ∩ {2, a, b}
= {2}
(d) (A ∪ B) ∪ C
Solution:
(A ∪ B) = {1, 2, 3, a}
Therefore,
(A ∪ B) ∪ C = {1, 2, 3, a} ∪ {a, b}
= {1, 2, 3, a, b}
(e) (A ∩ B) ∪ (B ∩ C)
Solution:
A ∩ B = {1, 2, 3} ∩ {2, a} = {2}
B ∩ C = {2, a} ∩ {a, b} = {a}
Therefore,
(A ∩ B) ∪ (B ∩ C) = {2} ∪ {a}
= {2, a}
9. If U = {1, 2, 3, 4, 5, 6, 7}, A = {1, 2, 5}, B = {2, 4, 7}, and C = {4, 5, 6}, justify the correctness of the following relations:
(a) A ∩ B = B ∩ A
Solution: Left-hand side = A ∩ B
= {1, 2, 5} ∩ {2, 4, 7}
= {2}
Right-hand side = B ∩ A
= {2, 4, 7} ∩ {1, 2, 5}
= {2}
Since Left-hand side = Right-hand side,
Therefore, the relation is true. (Verified)
(b) (A ∩ B)′ = A′ ∪ B′
Solution: Here,
A ∩ B = {1, 2, 5} ∩ {2, 4, 7} = {2}
Left-hand side = (A ∩ B)′
= U – (A ∩ B)
= {1, 2, 3, 4, 5, 6, 7} – {2}
= {1, 3, 4, 5, 6, 7}
Again,
A′ = U – A
= {1, 2, 3, 4, 5, 6, 7} – {1, 2, 5}
= {3, 4, 6, 7}
B′ = U – B
= {1, 2, 3, 4, 5, 6, 7} – {2, 4, 7}
= {1, 3, 5, 6}
Right-hand side = A′ ∪ B′
= {3, 4, 6, 7} ∪ {1, 3, 5, 6}
= {1, 3, 4, 5, 6, 7}
Since Left-hand side = Right-hand side,
Therefore, the relation is true. (Verified)
(c) (A ∪ C)′ = A′ ∩ C′
Solution: Here,
A ∪ C = {1, 2, 5} ∪ {4, 5, 6} = {1, 2, 4, 5, 6}
Left-hand side = (A ∪ C)′
= U – (A ∪ C)
= {1, 2, 3, 4, 5, 6, 7} – {1, 2, 4, 5, 6}
= {3, 7}
Again,
A′ = U – A = {3, 4, 6, 7}
C′ = U – C
= {1, 2, 3, 4, 5, 6, 7} – {4, 5, 6}
= {1, 2, 3, 7}
Right-hand side = A′ ∩ C′
= {3, 4, 6, 7} ∩ {1, 2, 3, 7}
= {3, 7}
Since Left-hand side = Right-hand side,
Therefore, the relation is true. (Verified)
10. If P and Q are the sets of all factors of 21 and 35 respectively, find P ∪ Q.
Solution:
Factors of 21 are: 1, 3, 7, 21
Therefore, P = {1, 3, 7, 21}
Factors of 35 are: 1, 5, 7, 35
Therefore, Q = {1, 5, 7, 35}
P ∪ Q = {1, 3, 7, 21} ∪ {1, 5, 7, 35}
= {1, 3, 5, 7, 21, 35} (Answer)
11. Answer the questions based on the Venn diagram:
Obtaining elements from the Venn diagram:
U = {3, 4, 5, 6, 7, 8, 9, 10}
A = {3, 4, 5, 6}
B = {4, 5, 8, 9}
C = {5, 6, 7, 8}
(a) Write set A in set builder method.
Solution:
In roster method, A = {3, 4, 5, 6}
The elements of set A are natural numbers that are greater than 2 and less than 7 (or greater than or equal to 3 and less than or equal to 6).
Required set: {x : x is a natural number and 2 < x < 7} (Answer)
(b) Express A, B, and C in tabular method, and find A ∩ C and A ∪ B.
Solution:
In roster method:
A = {3, 4, 5, 6}
B = {4, 5, 8, 9}
C = {5, 6, 7, 8}
Now,
A ∩ C = {3, 4, 5, 6} ∩ {5, 6, 7, 8}
= {5, 6} (Answer)
A ∪ B = {3, 4, 5, 6} ∪ {4, 5, 8, 9}
= {3, 4, 5, 6, 8, 9} (Answer)
(c) Prove that (A ∪ B)′ = A′ ∩ B′
Solution:
From the Venn diagram:
U = {3, 4, 5, 6, 7, 8, 9, 10}
A = {3, 4, 5, 6}
B = {4, 5, 8, 9}
Here,
A ∪ B = {3, 4, 5, 6, 8, 9}
Left-hand side = (A ∪ B)′
= U – (A ∪ B)
= {3, 4, 5, 6, 7, 8, 9, 10} – {3, 4, 5, 6, 8, 9}
= {7, 10}
Again,
A′ = U – A
= {3, 4, 5, 6, 7, 8, 9, 10} – {3, 4, 5, 6}
= {7, 8, 9, 10}
B′ = U – B
= {3, 4, 5, 6, 7, 8, 9, 10} – {4, 5, 8, 9}
= {3, 6, 7, 10}
Right-hand side = A′ ∩ B′
= {7, 8, 9, 10} ∩ {3, 6, 7, 10}
= {7, 10}
Since Left-hand side = Right-hand side,
Therefore, (A ∪ B)′ = A′ ∩ B′ (Proved)
12. The sets of natural integers by which the numbers 346 and 556 are divided with remainder 31 in each case are A and B.
(a) Express set A in set builders form.
Solution:
The natural numbers that leave a remainder of 31 when dividing 346 are numbers greater than 31 and factors of (346 – 31) or 315.
Factors of 315:
315 = 1 × 315
= 3 × 105
= 5 × 63
= 7 × 45
= 9 × 35
= 15 × 21
Among the factors of 315, the numbers greater than 31 are:
35, 45, 63, 105, 315.
Answer:
A = {35, 45, 63, 105, 315}
(b) Find A ∩ B.
Solution:
From ‘(a)’, A = {35, 45, 63, 105, 315}
Again, the natural numbers that leave a remainder of 31 when dividing 556 are numbers greater than 31 and factors of (556 – 31) or 525.
Factors of 525:
525 = 1 × 525
= 3 × 175
= 5 × 105
= 7 × 75
= 15 × 35
= 21 × 25
Among the factors of 525, the numbers greater than 31 are:
35, 75, 105, 175, 525.
Therefore,
B = {35, 75, 105, 175, 525}
Now,
A ∩ B = {35, 45, 63, 105, 315} ∩ {35, 75, 105, 175, 525}
= {35, 105}
Answer:
{35, 105}
(c) Show A ∩ B in Venn diagram and write the subsets of A ∩ B.
Solution:
From ‘(b)’, A ∩ B = {35, 105}
Venn Diagram:
Plaintext
┌───────────────────────────────────────────────┐
│ U │
│ ┌─────────────────┐ ┌─────────────────┐ │
│ │ A │ │ B │ │
│ │ │ │ │ │
│ │ 45, 63, │35 │ 75, 175, │ │
│ │ 315 │105│ 525 │ │
│ │ │ │ │ │
│ └─────────────────┘ └─────────────────┘ │
└───────────────────────────────────────────────┘
(Explanation: The intersection of circles A and B contains 35 and 105, which represents A ∩ B.)
Subsets of A ∩ B:
The subsets of the set A ∩ B = {35, 105} are:
{35, 105}, {35}, {105}, ∅
Answer:
{35, 105}, {35}, {105}, ∅
Sample Questions (Multiple Choice)
1. How many systems are there to express set?
a) 1
b) 2
c) 3
d) 4
Solution: Sets are mainly expressed in 2 methods (Roster method and Set-builder method).
Correct answer: b) 2
2. S = {x : x is an even number and 1 ≤ x ≤ 7} which of the following is correct in Tabular set system?
a) {2, 3, 4}
b) {2, 4, 6}
c) {1, 3, 5}
d) {3, 5, 7}
Solution: The even natural numbers between 1 and 7 are: 2, 4, 6.
Correct answer: b) {2, 4, 6}
3. If A = {2, 3, 5}
i. A = {x ∈ N : 1 < x < 6 and x is a prime number}
ii. A = {x ∈ N : 2 ≤ x < 7 and x is a prime number}
iii. A = {x ∈ N : 2 ≤ x ≤ 5 and x is a prime number}
Which one of the following is correct?
a) i and ii
b) i and iii
c) ii and iii
d) i, ii, and iii
Solution:
For i: Prime numbers greater than 1 and less than 6 are 2, 3, 5. (Correct)
For ii: Prime numbers greater than or equal to 2 and less than 7 are 2, 3, 5. (Correct)
For iii: Prime numbers greater than or equal to 2 and less than or equal to 5 are 2, 3, 5. (Correct)
Correct answer: d) i, ii, and iii
Answer questions 4 and 5 in light of the information below:
U = {2, 3, 5, 7}, A = {2, 5}, B = {3, 5, 7}
4. Which one is Aᶜ?
a) {2, 5}
b) {3, 5}
c) {3, 7}
d) {2, 7}
Solution:
Aᶜ = U – A
= {2, 3, 5, 7} – {2, 5}
= {3, 7}
Correct answer: c) {3, 7}
5. Which one is A ∩ Bᶜ?
a) {2}
b) {5}
c) {2, 5}
d) {3, 7}
Solution:
Bᶜ = U – B
= {2, 3, 5, 7} – {3, 5, 7}
= {2}
Now,
A ∩ Bᶜ
= {2, 5} ∩ {2}
= {2}
Correct answer: a) {2}
Creative Questions
6. In a hostel, 65% of students like fish, 55% of students like meat, and 40% of students like both dishes.
(a) Write the subsets of A = {x : x is an even natural number and x² ≤ 16}.
Solution:
Even natural numbers are: 2, 4, 6, 8…
When x = 2, x² = 4 ≤ 16
When x = 4, x² = 16 ≤ 16
When x = 6, x² = 36 (which is not less than or equal to 16)
Therefore,
A = {2, 4}
Subsets of set A are: {2, 4}, {2}, {4}, ∅
(Answer)
(b) Find out the percentage of students who dislike both dishes.
Solution:
Let, total number of students = 100%
Students who like at least one food (fish or meat)
= (like fish) + (like meat) – (like both)
= (65 + 55 – 40)%
= (120 – 40)%
= 80%
Therefore, students who do not like either food = (100 – 80)%
= 20%
Answer:
20% students
(c) Find out the intersection set of the sets of factors of those students who like only one dish.
Solution:
Like only fish = (65 – 40)% = 25%
Like only meat = (55 – 40)% = 15%
Let,
Set of factors of 25 = M
Factors of 25 are: 1, 5, 25
Therefore, M = {1, 5, 25}
Set of factors of 15 = N
Factors of 15 are: 1, 3, 5, 15
Therefore, N = {1, 3, 5, 15}
Required intersection set (M ∩ N):
M ∩ N = {1, 5, 25} ∩ {1, 3, 5, 15}
= {1, 5} (Answer)
7. Universal set U = {1, 2, 3, 4, 5, 6, 7} has three subsets:
A = {x : x² – 6x + 8 = 0}
B = {x : x is an odd number and 1 ≤ x ≤ 5}
C = {x : x is a prime number and 2 ≤ x ≤ 6}
(a) Express P = {3, 5, 7, 11} in set-builder notation.
Solution:
The elements 3, 5, 7, 11 are prime numbers greater than 2 and less than 13.
Required set: {x : x is a prime number and 2 < x < 13} (Answer)
(b) Write all the subsets of set A.
Solution:
Given, x² – 6x + 8 = 0
or, x² – 4x – 2x + 8 = 0
or, x(x – 4) – 2(x – 4) = 0
or, (x – 4)(x – 2) = 0
Either,
x – 4 = 0
or, x = 4
Or,
x – 2 = 0
or, x = 2
Therefore,
A = {2, 4}
All subsets of set A are: {2, 4}, {2}, {4}, ∅
(Answer)
(c) Find the complement of (B ∪ C)ᶜ.
Solution:
From the stem:
U = {1, 2, 3, 4, 5, 6, 7}
B = {x : x is an odd number and 1 ≤ x ≤ 5}
= {1, 3, 5}
C = {x : x is a prime number and 2 ≤ x ≤ 6}
= {2, 3, 5}
Now,
B ∪ C = {1, 3, 5} ∪ {2, 3, 5}
= {1, 2, 3, 5}
Therefore,
(B ∪ C)ᶜ = U – (B ∪ C)
= {1, 2, 3, 4, 5, 6, 7} – {1, 2, 3, 5}
= {4, 6, 7} (Answer)
Short-Answer Questions
8. (a) If A = {x ∈ N : x is even, 1 ≤ x ≤ 6}, write all proper subsets of A.
Solution:
The even natural numbers between 1 and 6 are: 2, 4, 6
Therefore, A = {2, 4, 6}
Excluding the set itself {2, 4, 6}, all other subsets are its proper subsets.
Required proper subsets:
{2, 4}, {2, 6}, {4, 6}, {2}, {4}, {6}, ∅
(Answer)
(b) If U = {1, 2, 3, 4, 5, 6} and B = {2, 4, 6}, find the complement of Bᶜ.
Solution:
Bᶜ = U – B
= {1, 2, 3, 4, 5, 6} – {2, 4, 6}
= {1, 3, 5} (Answer)
(c) Express B = {x ∈ N : x is a prime factor of 24} in tabular method.
Solution:
All factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Among these, the prime numbers are: 2, 3
B = {2, 3} (Answer)
(d) If P = {x : x is a factor of 6} and Q = {x : x is a factor of 8}, find P ∪ Q.
Solution:
Factors of 6: 1, 2, 3, 6
Therefore, P = {1, 2, 3, 6}
Factors of 8: 1, 2, 4, 8
Therefore, Q = {1, 2, 4, 8}
Required P ∪ Q:
P ∪ Q = {1, 2, 3, 6} ∪ {1, 2, 4, 8}
= {1, 2, 3, 4, 6, 8} (Answer)