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NCERT Class 11 Mathematics Chapter 1 Sets
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Sets
Chapter – 1
Exercise 1.1 |
Q1. Which of the following are sets? Justify our answer.
(i) The collection of all months of a year beginning with the letter J.
Ans: The collection of all months of a year beginning with the letter J is a well-defined collection of objects because one can definitely identify a month that belongs to this collection.
Hence, this collection is a set.
(ii) The collection of ten most talented writers of India.
Ans: The collection of ten most talented writers of India is not a well-defined collection because the criteria for determining a writer’s talent may vary from person to person.
Hence, this collection is not a set.
(iii) A team of eleven best-cricket batsmen of the world.
Ans: A team of eleven best cricket batsmen of the world is not a well-defined collection because the criteria for determining a batsman’s talent may vary from person to person.
Hence, this collection is not a set.
(iv) The collection of all boys in your class.
Ans: The collection of all boys in your class is a well-defined collection because you can definitely identify a boy who belongs to this collection.
Hence, this collection is a set.
(v) The collection of all natural numbers less than 100.
Ans: The collection of all natural numbers less than 100 is a well-defined collection because one can definitely identify a number that belongs to this collection.
Hence, this collection is a set.
(vi) A collection of novels written by the writer Munshi Prem Chand.
Ans: A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify a book that belongs to this collection.
Hence, this collection is a set.
(vii) The collection of all even integers.
Ans: The collection of all even integers is a well-defined collection because one can definitely identify an even integer that belongs to this collection.
Hence, this collection is a set.
(viii) The collection of questions in this Chapter.
Ans: The collection of questions in this chapter is a well-defined collection because one can definitely identify a question that belongs to this chapter.
Hence, this collection is a set.
(ix) A collection of most dangerous animals of the world.
Ans: The collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal can vary from person to person.
Hence, this collection is not a set.
Q2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 5…A.
Ans: 5 ∈ A, as 5 is the 5ᵗʰ element in set A.
(ii) 8…A.
Ans: 8 ∉ A, as 8 is not present in set A.
(iii) 0…A.
Ans: 0 ∉ A, as o is not present in set A.
(iv) 4…Α.
Ans: 4 ∈ A, as 4 is the 4ᵗʰ element in set A.
(ν) 2…A.
Ans: 2 ∈ A, as 2 is the 2nd element in set A.
(vi) 10…A
Ans: 10 ∉ A as 10 is not present in set A.
3. Write the following sets in roster form:
(i) A = {x: x is an integer and – 3 ≤ x < 7}.
Ans: A = {x: x is an integer and – 3 < x < 7}
The elements of this set are – 2, – 1, 0, 1, 2,
3, 4, 5, and 6 only.
Therefore, the given set can be written in roster form as
A = {- 2, – 1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {x: x is a natural number less than 6}.
Ans: B = {x: x is a natural number less than 6}.
The elements of this set are 1, 2, 3, 4, and 5 only.
Therefore, the given set can be written in roster form as
B = {1, 2, 3, 4, 5}.
(iii) C = {x: x is a two-digit natural number such that the sum of its digits is 8}.
Ans: C = {x: x is a two-digit natural number such that the sum of its digits is 8}
The elements of this set are 17, 26, 35, 44, 53, 62, 71, and 80 only.
Therefore, this set can be written in roster form as
C= {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {x: x is a prime number which is divisor of 60}.
Ans: D = {x: x is a prime number which is divisor of 60}.
2 | 60 |
2 | 30 |
3 | 15 |
5 |
60 = 2 x 2 x 3 x 5
The elements of this set are 2, 3, and 5 only.
Therefore, this set can be written in roster form as D = {2, 3, 5}.
(v) E = The set of all letters in the word TRIGONOMETRY.
Ans: E = The set of all letters in the word TRIGONOMETRY
There are 12 letters in the word TRIGONOMETRY, out of which letters T, R, and O are repeated.
Therefore, this set can be written in roster form as
E = {T, R, I, G, O, N, M, E, Y}
(vi) F= The set of all letters in the word BETTER.
Ans: F = The set of all letters in the word BETTER
There are 6 letters in the word BETTER, out of which letters E and T are repeated.
Therefore, this set can be written in roster form as
F = {B, E, T, R}
Q4. Write the following sets in the set – builder form:
(i) (3, 6, 9, 12}.
Ans: {3, 6, 9, 12}
= {x : x = 3n, n∈N and 1≤ n ≤4}
(ii) {2, 4, 8, 16, 32}.
Ans: {2, 4, 8, 16, 32}
It can be seen that 2 = 2¹,4 = 2²,8 = 2³,16 = 2⁴ and 32 = 2⁵.
∴ {2,4,8,16,32} = {x: x = 2ⁿ, n∈ N and 1≤ n ≤ 5}
(iii) {5, 25, 125, 625}.
Ans: {5, 25, 125, 625}
It can be seen that 5 = 5¹, 25 = 5²,125 = 5³ and 625 = 5⁴.
∴ {5,25,125,625} = {x: x = 5ⁿ, n∈ N and 1 ≤ n ≤ 4}
(iv) {2, 4, 6 …}.
Ans: {2, 4 ,6…}
It is a set of all even natural numbers.
∴ {2, 4 ,6…} = {x: x is an even natural number}
(v) {1, 4, 9 … 100}.
Ans: {1, 4 ,9…100}
It can be seen that 1 = 1²,4 = 2²,9 = 3²…..100 = 10².
∴ {1,4,9…100} = {x: x = n², n∈N and 1≤ n ≤ = 10}
Q5. List all the elements of the following sets:
(i) A = {x : x is an odd natural number}.
Ans: A = {x : x is an odd natural number} ={1, 3, 5, 7 ,9…}
(ii) B = {x: x is an integer,}
Ans: B = {x: x is an integer; It
can be seen that
∴ = {0, 1, 2, 3, 4}
(iii) C = {x: x is an x² ≤ 4} integer,
Ans: C = {x:x is an integer; x² ≤4 It can be seen that
(-1)² = 1 ≤ 4
(- 2)² = 4 ≤ 4
(- 3)² = 9 > 4
0² = 0 ≤ 4
1² = 1 ≤ 4
2² = 4 ≤ 4
3² = 9 > 4
∴ C = {- 2,- 1,0,1,2}
(iv) D = {x : x is a letter in the word “LOYAL”}
And: D = (x : x is a letter in the word “LOYAL”) = {L, O, Y, A}
(v) E={x: x is a month of a year not having 31 days}
Ans: E = {x : x is a month of a year not having 31 days}
= {February, April, June, September, November}
(vi) F={x: x is a consonant in the English alphabet which proceeds k}.
Ans: F = {x : x is a consonant in the English alphabet which precedes k}
= {b, c, d, f, g, h, j}
Q6. Match each of the set on the left in the roster form with the same set on the right described in set – builder form:
(i) {1, 2, 3, 6} (a) {x: x is a prime number and a divisor of 6}.
Ans: All the elements of this set are natural numbers as well as the divisors of 6. Therefore, (i) matches with (c).
(ii) {2, 3} (b) {x: x is an odd natural number less than 10}
Ans: It can be seen that 2 and 3 are prime numbers. They are also the divisors of 6.
Therefore, (ii) matches with (a).
(iii) {M, A,T, H, E, I,C, S} (c) {x: x is natural number and divisor of 6}
Ans: All the elements of this set are letters of the word MATHEMATICS. Therefore, (iii) matches with (d).
(iv) {1, 3, 5, 7, 9} (d) {x: x is a letter of the word MATHEMATICS}
Ans: All the elements of this set are odd natural numbers less than 10. Therefore, (iv) matches with (b).
Exercise 1.2 |
Q1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 2.
Ans: A set of odd natural numbers divisible by 2 is a null set because no odd number is divisible by 2.
(ii) Set of even prime numbers.
Ans: A set of even prime numbers is not a null set because 2 is an even prime number.
(iii) {x : x is a natural numbers, x < 5 and x >7}
Ans: {x: x is a natural number, x < 5 and x > 7} is a null set because a number cannot be simultaneously less than 5 and greater than 7.
(iv) {y : y is a point common to any two parallel lines}.
Ans: {y: y is a point common to any two parallel lines} is a null set because parallel lines do not intersect. Hence, they have no common point.
Q2. Which of the following sets are finite or infinite.
(i) The set of months of a year.
Ans: The set of months of a year is a finite set because it has 12 elements.
(ii) {1, 2 ,3…}
And: {1, 2, 3 …} is an infinite set as it has an infinite number of natural numbers.
(iii) {1, 2 ,3…99,100}
Ans: {1, 2, 3 …99, 100} is a finite set because the numbers from 1 to 100 are finite in number.
(iv) The set of positive integers greater than 100.
And: The set of positive integers greater than 100 is an infinite set because positive integers greater than 100 are infinite in number.
(v) The set of prime numbers less than 99.
Ans: The set of prime numbers less than 99 is a finite set because prime numbers less than 99 are finite in number
Q3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis.
And: The set of lines which are parallel to the x-axis is an infinite set because lines parallel to the x-axis are infinite in number.
(ii) The set of letters in the English alphabet.
Ans: The set of letters in the English alphabet is a finite set because it has 26 elements.
(iii) The set of numbers which are multiple of 5.
Ans: The set of numbers which are multiple of 5 is an infinite set because multiples of 5 are infinite in number.
(iv) The set of animals living on the earth.
Ans: The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number).
(v) The set of circles passing through the origin (0, 0).
Ans: The set of circles passing through the origin (0, 0) is an infinite set because an infinite number of circles can pass through the origin.
Q4. In the following, state whether A = B or not:
(i) A = {a, b, c, d}; B = {d, c, b, a}.
Ans: A = {a, b, c, d}; B = {d, c, b, a}
The order in which the elements of a set are listed is not significant.
A = B
(ii) A = {4, 8, 12, 16}; B = {8, 4, 16, 18}.
Ans: A = {4, 8, 12, 16} B = {8, 4, 16, 18} It can be seen that 12 ∈ A but 12 ∉ B.
∴ A ≠ B
(iii) A = {2, 4, 6, 8, 10} B = {x: x is positive even Integer and x ≤ 10}.
Ans: A = {2, 4, 6, 8, 10}
B = {x: x is a positive even integer and x ≤ 10}
= {2, 4, 6, 8, 10}
∴ A=B
(iv) A = {x: x is a multiple of 10}, B = {10, 15, 20, 25, 30…}.
Ans: A = {x: x is a multiple of 10}
B = {10, 15, 20, 25, 30 …}
It can be seen that 15 ∈ B but 15 ∉ A.
∴ A ≠ B
Q5. Are the following pair of sets equal? Give reasons.
(i) A = {2, 3};B = {x: x is solution of x² + 5x + 6 = 0}.
Ans: A = {2, 3}; B = {x: x is a solution of x² + 5x + 6 = 0}
The equation x² + 5x + 6 = 0 can be solved as: x(x + 3) + 2(x + 3) = 0(x + 2)(x + 3) = 0
x = – 2 or x = – 3
∴ A = {2, 3}; B = {- 2, – 3}
∴ A ≠ B
(ii) A = {x: x is a letter in the word FOLLOW};
B = {y: y is a letter in the word WOLF}.
Ans: A = {x: x is a letter in the word FOLLOW} = {F, O, L, W}
B = {y: y is a letter in the word WOLF} = {W, O, L, F}.
The order in which the elements of a set are listed is not significant
∴ A = B
Q6. From the sets given below, select equal sets:
A = {2, 4, 8, 12}, B = {1, 2, 3, 4}, C = {4, 8, 12, 14},D = {3, 1, 4, 2},E = {- 1, 1}, F = {0, a} G = {1, – 1}, H = {0, 1}.
Ans: A = {2, 4, 8, 12}
n(A) = 4
B = {1, 2, 3, 4}
n(B) = 4
C = {4, 8, 12, 14}
n(C) = 4
D = {3, 1, 4, 2}
n(D) = 4
E = {- 1, 1}
n(E) = 2
F = {0, a}
n(F) = 2
G = {1, – 1}
n(G) = 2
H = {0, 1}
n(H) = 2
Number of elements in A,B,C,D are same i.e. all have 4 elements. So, they are comparable.
Now, we see B and D has same elements.
So, B and D are equal sets.
Similarly, E,F,G, H are comparable as they all have same number of elements
i.e. 2.
Clearly, E and G has same elements.
So, E and G are equal sets.
Exercise 1.3 |
1. Make correct statements by filling in the symbols ⊂ or ⊄ in the blank spaces:
(i) {2, 3, 4}…..{1,2,3,4,5}.
Ans: {2, 3, 4} ⊂ {1,2,3,4,5}.
(ii) (a,b,c}…… {b,c,d}.
Ans: (a,b,c} ⊄ {b,c,d}.
(iii) {x : x is a student of Class XI of your school}….{x : x student of your school}.
Ans: {x: x is a student of class XI of your school} ⊂ {x: x is student of your school}
(iv) {x : x is a circle in the plane}…….{x : x is a circle in the same plane with radius 1 unit}.
Ans: {x: x is a circle in the plane} ⊄ {x: x is a circle in the same plane with radius 1 unit}.
(v) {x : x is a triangle in a plane}………{x : x is a rectangle in the plane}.
Ans: {x : x is a triangle in a plane} ⊄ {x : x is a rectangle in the plane}
(Vi) {x : x is an equilateral triangle in a plane}….{x:x is an triangle in the same plane}.
Ans: {x : x is an equilateral triangle in a plane}⊂ {x: x in a triangle in the same plane}
(vii) {x : x is an even natural number}…..{x : x is an integer}.
Ans: {x : x is an even natural number} ⊂ {x : x is an integer}.
Q2: Examine whether the following statements are true or false:
(i) {a, b} ⊄ {b, c, a}.
Ans: False. Each element of {a, b} is also an element of {b, c, a}.
(ii) {a, e} ⊂ {x : x is a vowel in the English alphabet}.
Ans: True. a, e are two vowels of the English alphabet.
(iii) {1, 2, 3} ⊂ {1, 3, 5}.
Ans: False. 2 ∈ {1, 2, 3}; however, 2 ∉ {1, 3, 5}.
(iv) {a} ⊂ {a. b, c}.
Ans: True. Each element of {a} is also an element of {a, b, c}.
(v) {a} ∈ {a, b, c}.
Ans: False. The elements of {a, b, c} are a, b, c. Therefore, {a} ⊂ {a, b, c}.
(vi) {x: x is an even natural number less than 6} ⊂ {x: x is a natural number which divides 36}.
Ans: True. {x : x is an even natural number less than 6} = {2, 4} {x : x is a natural number which divides 36}={1, 2, 3, 4, 6, 9, 12, 18, 36}.
3. Let A = {1,2, {3,4},5}. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A.
Ans: The statement {3, 4} ⊂ A is incorrect because 3 ∈ {3, 4}; however, 3 ∉ A.
(ii) {3,4} ∈ A.
Ans: The statement {3, 4} A is correct because {3, 4} is an element of A.
(iii) {{3,4}} ⊂ A.
Ans: The statement {{3, 4}} ⊂ A is correct because {3, 4} ∈ {{3, 4}} and {3, 4} ∈ A.
(iv) 1 ∈ A.
Ans: The statement 1∈ A is correct because 1 is an element of A.
(v) 1 ⊂ A.
Ans: The statement 1⊂ A is incorrect because an element of a set can never be a subset of itself.
(vi) {1,2,5} ⊂ A.
Ans: The statement {1, 2, 5} ⊂ A is correct
because each element of {1, 2, 5 } is also an element of A.
(vii) {1, 2, 5} ∈ A.
Ans: The statement {1, 2, 5} ∈ A is incorrect
because {1, 2, 5} is not an element of A.
(viii) {1, 2, 3} ⊂ A.
Ans: The statement {1, 2, 3} ⊂ A is incorrect because 3 ∈ {1, 2, 3}; however, 3 ∉ A.
(ix) Φ ∈ Α.
Ans: The statement Φ ∈ A is incorrect because
Φ is not an element of A.
(Χ) Φ ⊂ Α.
Ans: The statement Φ ⊂ A is correct because
Φ is a subset of every set.
(xi) {Φ} ⊂ Α.
Ans: The statement {Φ} ⊂ A is incorrect because Φ ∈ {Φ}; however,Φ ∈ A.
Q4. Write down all the subsets of the following sets:
(i) {a}.
Ans: The subsets of {a} are Φ and {a}.
(ii) {a, b}.
Ans: The subsets of {a, b} areΦ , {a}, {b}, and {a, b}.
(iii) {1, 2, 3}.
Ans: The subsets of {1, 2, 3} are Φ, {1}, {2}, {3}, {1, 2} , {2, 3}, {1, 3} and {1, 2, 3}.
(iv) Φ.
Ans: The only subset of Φ is Φ.
Q5. Write the following as intervals:
(i) {x : x ∈ R, – 4 < x ≤ 6}
Ans: {x : x ∈R, – 4 < x ≤ 6} = (- 4, 6]
(ii) {x : x∈R, – 12 < x < – 10}
Ans: {x : x ∈R,-12 < x < – 10} = (- 12, – 10).
(iii) {x : x ∈ R, 0 ≤ x < 7}
Ans: {x : x ∈ R, 0 ≤ x < 7} = [0,7)
(iv) { x : x ∈ R, 3 ≤ x ≤ 4}
Ans: {x : x ∈ R, 3 ≤ x ≤ 4} = [3, 4]
Q6. Write the following intervals in set-builder form:
(i) (-3,0)
Ans: (-3, 0) = {x: x ∈ R, -3 < x < 0}.
(ii) [6, 12]
Ans: [6, 12] = {x: x ∈R, 6 ≤ x ≤ 12}.
(iii) (6, 12]
Ans: (6, 12] = {x: x ∈ R, 6 < x ≤ 12}.
(iv) [-23, 5)
Ans: [-23, 5) = {x: x ∈ R, -23 ≤ x < 5}
Q7. What universal set (s) would you propose for each of the following:
(i) The set of right triangles.
Ans: For the set of right triangles, the universal set can be the set of triangles or the set of polygons.
(ii) The set of isosceles triangles.
Ans: For the set of isosceles triangles, the universal set can be the set of triangles or the set of polygons or the set of two-dimensional figures.
Q8. Given the sets A = {1, 3, 5} B = {2, 4, 6} and C = {0, 2, 4, 6, 8} which of the following may be considered as universals set (s) for all the three sets A, B and C.
(i) {0, 1, 2, 3, 4, 5, 6}.
Ans: It can be seen that A ⊂ {0, 1, 2, 3, 4, 5, 6}
B ⊂ {0, 1, 2, 3, 4, 5, 6}
However, C /⊂ {0, 1, 2, 3, 4, 5, 6}
Therefore, the set {0, 1, 2, 3, 4, 5, 6} cannot be the universal set for the sets A B and C.
(ii) Ф
Ans: А /⊂Ф, В /⊂ Ф, С /⊂ Ф
Therefore, Ф cannot be the universal set for the sets A, B, and C.
(iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Ans: A ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
B ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
C ⊂ {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Therefore, the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is the universal set for the sets A, B and C.
(iv) {1, 2, 3, 4, 5, 6, 7, 8}
Ans: A ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
B ⊂ {1, 2, 3, 4, 5, 6, 7, 8}
However, C /⊂ {1, 2, 3, 4, 5, 6, 7, 8}
Therefore, the set {1, 2, 3, 4, 5, 6, 7, 8} cannot be the universal set for the sets A, B, and C.
Exercise 1.4 |
Q1. Find the union of each of the following pairs of sets:
(i) X = {1, 3, 5} Y = {1, 2, 3}
Ans: X = {1, 3, 5} Y = {1, 2, 3}
X ∪ Y = {1, 2, 3, 5}.
(ii) A = {a, e, i, o, u} B = {a, b, c}
Ans: A = {a, e, i, o, u} B = {a, b, c}
A ∪ B = {a, b, c, e, i, o, u}.
(iii) A = {x : x is a natural number and multiple of 3}.
B ={x : x is a natural number less than 6}
Ans: A = {x : x is a natural number and multiple of 3} = {3, 6, 9 …}
As B = {x : x is a natural number less than 6} = {1, 2, 3, 4, 5, 6}
AUB = {1, 2, 4, 5, 3, 6, 9, 12 …}
∴ A U B= {x : x = 1, 2, 4, 5 or a multiple of 3}.
(iv) A = {x : x is a natural number and l < x ≤ 6}
B = {x : x is a natural number and 6 < x < 10}
Ans: A = {x : x is a natural number and 1 < x ≤6}
A = {2, 3, 4, 5, 6}
B = {x : x is a natural number and 6 < x ≤10}
B = {7, 8, 9}
A U B = {x: x∈ N and 1 < x < 10}
(v) A = {1, 2, 3},B = Ф
Ans: A = {1, 2, 3} , Β = Φ
A ∪ B = {1, 2, 3}
Q2. Let A = {a,b}, B = {a, b, c}. Is A ⊂ B? What is A ∪ B?
Ans: Here, A = {a, b} and B = {a, b, c}
Since, every element of set A is in set B. So, A ⊂ B
A ∪ B = {a, b, c} = B
Q3. If A and B are two sets such that A ⊂ B, then what is A ∪ B?
Ans: Given that A ⊂ B, i.e., B will contain all the elements of A.
∴ A U B = B
Q4. If A = {1, 2, 3, 4} B = {3, 4, 5, 6} C = {5, 6, 7, 8} and D = {7, 8, 9, 10}; find :
(i) A U B.
Ans: AB = {1, 2, 3, 4, 5, 6}
(ii) A U C.
Ans: A U C = {1, 2, 3, 4, 5, 6,7,8}
(iii) B U C.
Ans: B U C = {3, 4, 5, 6, 7, 8}
(iv) B U D.
Ans: B U D = {3, 4, 5, 6, 7, 8, 9, 10}
(v) A U B U C.
Ans: A U B U C = {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A U B U D.
Ans: A U B U D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B U C U D.
Ans: B U C U D = {3, 4, 5, 6, 7, 8, 9, 10}
Q5. Find the intersection of each pair of sets of question 1 above.
(i) X = {1, 3, 5} Y = {1, 2, 3}.
Ans: X = {1, 3, 5} Y = {1, 2, 3}
X ∩ Y = {1, 3}.
(ii) A = {a, e, i, o, u} B = {a, b, c}
Ans: A = {a, e, i, o, u} B = {a, b, c}
A ∩ B = {a}.
(iii) A = {x: x is a natural number and multiple of 3}. B={ x: x is a natural number less than 6}
Ans: A ={x: xis a natural number and multiple of 3} =( 3, 6 ,9…}
B ={x: xis a natural number less than 6} = {1, 2, 3, 4, 5}
∴ A ∩ B = {3}.
(iv) A = {x : x is a natural number and 1 < x ≤6}
B = {x : x is a natural number and 6 < x ≤ 10}
Ans: A = {x : x is a natural number and 1 < x ≤6}
∴ A = {2, 3, 4, 5, 6}
B = {x : x is a natural number and 6 < x < 10}
∴ B = {7, 8, 9}
A ∩ B= {2, 3, 4, 5, 6} ∩ {7,8,9}
Hence, A ∩ B = ⍉
(v) A = {1, 2, 3}, B = ∅
Ans: A = {1, 2, 3} B = ∅
A ∩ B = ∅.
Q6: If A = {3, 5, 7, 9, 11} B = {7, 9, 11, 13} C = {11, 13, 15} and D = {15, 17}; find:
(i) A ∩ B.
Ans: A ∩ B = {7, 9, 11}.
(ii) B ∩ C.
Ans: B ∩ C = {11, 13}.
(iii) A ∩ C ∩ D.
Ans: A ∩ C ∩ D = { A ∩ C} ∩D = {11} ∩ {15, 17} = ∅.
(iv) A ∩ C.
Ans: A ∩ C = {11}.
(v) B ∩ D.
Ans: B ∩ D = ∅.
(vi) A ∩ (B ∪ C).
Ans: A ∩ (B ∪ C) =(A ∩ B) ∪ (A ∩ C)
= {7, 9 11} ∪ {11} = {7, 9, 11}.
(vii) A ∩ D.
Ans: A ∩ D = ∅.
(viii) A ∩ (B ∪ D).
Ans: A ∩ (B U D) = (A ∩ B) U (A ∩ D)
= {7,9,11} U ⍉ = {7, 9, 11}
(ix) (A ∩ B) ∩ (B U C).
Ans: (A ∩B) ∩ (B U C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15} = {7, 9, 11}.
(x) ( A U D) ∩ (B U C).
Ans: (A ∪ D) ∩ (B U C) = {3, 5, 7, 9, 11, 15, 17 } ∩ {7,9,11,13,15}
= {7, 9, 11, 15}.
Q7. If A = {x : x is a natural number}, B = {x: x is an even natural number}
C = {x: x is an odd natural number} and D = {x: x is a prime number}, find
(i) A ∩ B.
Ans: A ∩ B = {x:xis a even natural number} = B.
(ii) A ∩ C.
Ans: A ∩ C = {x:xis an odd natural number} = C.
(iii) A ∩ D.
Ans: A ∩ D = {x: x is a prime number} = D.
(iv) B ∩ C.
Ans: B ∩ C = ∅.
(v) B ∩ D.
Ans: B ∩ D = {2}.
(vi) C ∩ D.
Ans: C ∩ D = {x: x is odd prime number}.
Q8: Which of the following pairs of sets are disjoint:
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6}.
Ans: A = {1, 2, 3, 4}
B = { x : x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}
Now, A ∩ B = {1,2,3,4} ∩ {4, 5, 6} = {4}
Therefore, this pair of sets are not disjoint.
(ii) {a, e, i, o, u} and {c, d, e, f}.
Ans: {a, e, i, o, u} ∩ (c, d, e, f} = {e}
Therefore, {a, e, i, o, u} and (c, d, e, f} are not disjoint.
(iii) {x : x is an even integer} and {x : x is an odd integer}.
Ans: {x : x is an even integer} ∩ {x : x is an odd integer} = Ф
Therefore, this pair of sets is disjoint.
Q9. If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12, 16, 20}
C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20} find
(i) A – B.
Ans: A – B = {3, 6, 9, 15, 18, 21}.
(ii) A – C.
Ans: A – C = {3, 9, 15, 18, 21}.
(iii) A – D.
Ans: A – D = {3, 6, 9, 12, 18, 21}.
(iv) B – A.
Ans: B – A = {4, 8, 16, 20}.
(v) C – A.
Ans: C – A = {2, 4, 8, 10, 14, 16}.
(vi) D – A.
Ans: D – A = {5, 10, 20}.
(vii) B – C.
Ans: B – C = {20}.
(viii) B – D.
Ans: B – D = {4, 8, 12, 16}.
(ix) C – B.
Ans: C – B = {2, 6, 10, 14}.
(x) D – B.
Ans: D – B = {5, 10, 15}.
(xi) C – D.
Ans: C – D = {2, 4, 6, 8, 12, 14, 16}.
(xii) D – C.
Ans: D – C = {5, 15, 20}.
Q10. If X = {a, b, c, d} and Y = {f, b, d, g}, find
(i) X – Y.
Ans: X – Y = {a, c}.
(ii) Y – X.
Ans: Y – X = {f, g}.
(iii) X ∩ Y.
Ans: X ∩ Y = {b, d}.
Q11. If R is the set of real numbers and Q is the set of rational numbers, then what is R- Q?
Ans: R set of real numbers
Q. set of rational numbers
Therefore, R – Q is a set of irrational numbers.
Q12. State whether each of the following statement is true or false. Justify your answer.
(i) {2, 3, 4, 5} and {3, 6} are disjoint sets.
Ans: False
A = {2, 3, 4, 5} B = {3, 6}
⇒ A ∩ B = {3}
So, A and B are not disjoint sets.
(ii) {a, e, i, o, u} and {a, b, c, d} are disjoint sets.
Ans: False
A = {a, e, i, o, u}, B = {a, b, c, d}
A ∩ B = {a}
Hence, A and B are not disjoint sets.
(iii) {2, 6, 10, 14} and {3, 7, 11, 15} are disjoint sets.
Ans: True.
as {2,6,10,14} ∩ {3, 7, 11, 15} = Ф
(iv) {2, 6, 10} and {3, 7, 11} are disjoint sets.
Ans: True.
As {2,6,10 } ∩ {3, 7, 11} = Ф
Exercise 1.5 |
Q1. Let U = {1, 2, 3; 4, 5, 6, 7, 8, 9}, A = {1, 2, 3, 4}, B = {2, 4, 6, 8} and C = {3, 4, 5, 6}, Find
(i) A’.
Ans: A’ = {5, 6, 7, 8, 9}.
(ii) Β’.
Ans: B’ = {1, 3, 5, 7, 9}.
(iii) ( A U C)’.
Ans: A U C = {1, 2, 3, 4, 5, 6}
∴ (A U C)’ = {7, 8, 9}.
(iv) ( A U B)’.
Ans: A U B = {1, 2, 3, 4, 6, 8}
(A U B)’= {5, 7, 9}.
(v) (A’)’
Ans: (A’)’ = A = {1, 2, 3, 4}.
(vi) (B – C)’
Ans: B – C = {2, 8}
∴ (B – C)’ = {1, 3, 4, 5, 6, 7, 9}.
Q2. If U = {a, b, c, d, e, f, g, h} find the complements of the following sets:
(i) A = {a, b, c}.
Ans: A = {a, b, c}
A’ = {d, e, f, g, h}
(ii) B = {d, e, f, g}.
Ans: B = {d, e, f, g}
∴ B’ = {a,b,c,h}.
(iii) C = {a, c, e, g}.
Ans: C = {a, c, e, g}
∴ C’ = {b,d,f,h}
(iv) D = {f, g, h, a}.
Ans: D = {f, g, h, a}
∴ D’ = {b,c,d,e}
Q3. Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i) {x : x is an even natural number}.
Ans: {x: x is an even natural number}’ = {x: x is an odd natural number}.
(ii) {x: x is an odd natural number}.
Ans: {x: x is an odd natural number}’ = {x: x is an even natural number}.
(iii) {x: x is a positive multiple of 3}.
Ans:{x: x is a positive multiple of 3}’ = {x: x ∈ N
and x is not a multiple of 3}
(iv) {x: x is a prime number}.
Ans: {x: x is a prime number}’ = {x: x is a positive composite number and x = 1}
(v) {x: x is a natural number divisible by 3 and 5}.
Ans: {x: x is a natural number divisible by 3 and 5}’ = {x: x is a natural number that is not divisible by 3 or 5}
(vi) {x: x is a perfect square}.
Ans: {x: x is a perfect square}’ = {x: x ∈N and x is not a perfect square}
(vii) {x: x is perfect cube}.
Ans: {x: x is a perfect cube}’ = {x: x ∈N and x is not a perfect cube}.
(viii) {x: x + 5 = 8}.
Ans: {x : x + 5 = 8}’ = {x: x ∈N and x ≠ 3}
(ix) {x : 2x + 5 = 9}.
Ans: {x: 2x+5=9}’ = {x: x ∈N and x ≠ 2}.
(x) {x : x ≥ 7}.
Ans: {x: x ≥ 7}’ = {x: x ∈N and x <7}.
(xi) {x: x ∈ N and 2x + 1 > 10}.
Ans: {x: x ∈N and 2x + 1 > 10}’ = {x: x ∈N and x ≤ = 9/2}
Q4. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8} and B = {2, 3, 5, 7}. Verify that.
(i) (A U B)’ = A’ ∩ B’.
Ans: (A U B)’ = {2, 3, 4, 5, 6, 7, 8}’ = {1, 9}
A’ ∩ B’ = {1,3,5,7,9} ∩ (1, 4, 6, 8, 9) = {1, 9}
∴ (A U B)’ = A’ ∩ B’.
(ii) (A ∩ B) = (A’ U B).
Ans: (A ∩ B)’ = {2}’ = {1, 3, 4, 5, 6, 7, 8, 9}
A’ ∩ B’ = {1,3,5,7,9} ∩ {1, 4, 6, 8, 9} = {1, 3, 4, 5, 6, 7, 8, 9}
∴ (A ∩ B)’ =A’ U B’
Q5. Draw appropriate Venn diagram for each of the following:
(i) ( A ∪ B)’.
Ans:
(ii) A’ ∩ B’.
Ans:
(iii) ( A ∩ B)’.
Ans:
(iv) A’ ∪ B’.
Ans:
Q6. Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A’?
Ans: A’ is the set of all equilateral triangles.
Q7. Fill in the blanks to make each of the following a true statement:
(i) A ∪ A’=…
Ans: A ∪ A’= U.
(ii) ⍉’ ∩ A =…..
Ans: ⍉ ∩ A= U ∩ A = A
∴ ⍉ ∩ A = A
(iii) A ∩ A’=…
Ans: It can be clearly seen from the Venn diagram, there is no common portion between A and A’.
Hence, A ∩ A’ = ⍉
(iv) U’ ∩ A =…
And: U’ ∩ A = ⍉ ∩ A = ⍉
∴ U’ ∩ A = ⍉.
Miscellaneous Exercise on Chapter 1 |
Q1. Decide, among the following sets, which sets are subsets of one and another:
A = { x: x ∈ R and x satisfy x²- 8x + 12 = 0}
B = {2, 4, 6}, C = {2, 4, 6 ,8…}, D = {6}.
Ans: A = {x : x ∈ Rand x satisfiesx² – 8x + 12 = 0}
2 and 6 are the only solutions of x² – 8x + 12 = 0 .
∴ A = {2, 6}
B = {2, 4, 6}
C = {2, 4, 6, 8 ,….}
D = {6}
Clearly, D ⊂ A ⊂ B ⊂ C
Hence, A ⊂ B, A ⊂ C, B ⊂ C, D ⊂ A, D ⊂ B , D ⊂ C
Q2. In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
(i) If x ∈ A and A ∈ B, then x ∈ B.
And: False
Let A = {1, 2} and B = {1, {1, 2}, {3}}
2 ∈ {1, 2} and {1, 2} ∈ {{3}, 1, {1, 2}}
Now,
∴ Α ∈ Β
However, 2 /∈ {{3},1, {1, 2}}
(ii) If A ⊂ B and B ∈ C then A ∈ C.
Ans: False.
As A ⊂ B, B ∈ С
Let A = {2}, B = {0, 2}, and C = {1, {0,2},3}
However, A /∈C
(iii) If A ⊂ B and B ⊂ C, then A ⊂ C.
Ans: True
Let A ⊂ B and B ⊂ C.
Let x ∈ A
⇒ x ∈ B [∴ A ⊂ B]
⇒ x ∈ C [∵ B ⊂ C]
∴ A ⊂ C
Ans: False
As, A / ⊂B and B /⊂C
Let A = {1, 2}, B = {0, 6, 8}, and C = {0, 1, 2, 6, 9}
However, A ⊂ C.
Ans: False
Let A = {3, 5, 7} and B = {3, 4, 6}
Now, 5 ∈ A and A /⊂B
However, 5 /∈B
Ans: True .
Let A ⊂ B and x /∈B
To show: x / ∈ A
If possible, suppose x ∈ A
Then, x ∈ B , which is a contradiction as x /∈ B
∴x /∈A.
Q3. Let A, B, and C be the sets such that A ∪ B = A ∪ C and A ∩ B= A ∩C. Show that B = C.
Ans: Given that AUB = AUC
⇒ (AUB) ∩ C = (AUC) ∩C
⇒ (A∩C) U (B∩C) = C [∴(AUC)∩C = C ]
⇒ (A∩B) U (B∩C) = C ……….(1) [∴(A∩C) = A∩B]
Again AUB = AUC
(AUB) ∩ B = (AUC) ∩ B
B = (A∩B) U (C∩B)
= (A∩B) U (B∩C) ………..(2)
From 1 & 2 we get
B = C
Q4. Show that the following four conditions are equivalent:
(i) A ⊂ B.
Ans: First, we have to show that (i) ⇒ (ii).
(i) ⇒ (ii).
Let A ⊂ B
To show: A – B = ⍉
If possible, suppose A – B ≠ ⍉
This means that there exists x ∈ A, x ∉ B which is not possible as A ⊂ B .
∴ A – B = ⍉
∴ A ⊂ B ⇒ A – B = ⍉
(ii) A – B = ∅.
Ans: (ii) ⇒ (i).
Let A – B = ⍉
To show: A ⊂ B
Let x∈A
Clearly, x∈B because if x /∈B, then A – B ≠ ⍉
∴ A – B = ⍉ ⇒ A ⊂ B
Hence, (ii) ⇒ (i)
(iii) A∪ B = B.
Ans: (i) ⇒ (iii).
Let A ⊂ B
To show: A U B = B
Clearly, B ⊂ A U B
Let x∈A U B
⇒ x∈A or x∈Β
Case I: X∈A
⇒ x∈B [∵ A ⊂ B]
∴ A U B ⊂ B
Case II: X∈B
Then A U B ⊂ B
So, A U B = B
(iii) ⇒(i)
Conversely, let A U B = B
To show: A ⊂ B
Let x∈A
⇒ x∈A U B[∴ A ⊂ A U B]
⇒ x∈B [∵ A U B = B]
∴ A ⊂ B
Hence, (iii) ⇒ (i)
(iv) A ∩ B=A.
Ans: Now, we have to show that (i) ⇒ (iv).
Let A ⊂ B
Clearly A ∩ B ⊂ A
Let x∈A
We have to show that x∈A ∩ B
As A ⊂ B, X∈Β
∴ x∈A ∩ B
∴ A ⊂ A ∩ B
Hence, A = A ∩ B
Conversely, suppose A ∩ B = A
Let x∈A
⇒ X∈A ∩ B
⇒ x∈A and x∈B
⇒ x∈Β
∴ A ⊂ B
Hence, (i) ⇒ (iv).
Q5. Show that if A ⊂ B, then C – B ⊂ C -A.
Ans: Let A ⊂ B
To show: C- B ⊂ C – A
Let x ∈ C – B
⇒ x ∈ C and x /∈ B
⇒ x ∈ C and x /∈A(A ⊂ B)
⇒ x ∈ C – A
∴ C – B ⊂ C – A
Q6. Show that for any sets A and B, A=(A∩ B) U (A – B) and A∪ (B-A)= (A∪B).
Ans: (A∩B) u (A – B)
= (A∩B) U (Α∩Β΄)
= A ∩ (B U B’) (By distributive law)
= A ∩ U = A
Hence A = (A∩B) u (A – B)
Also A u (B-A)
= A U (B ∩ A’)
= (A U B) ∩ (A U A’) (By distributive law)
= (AUB) ∩ U
= A U B
Hence A u (B – A) = A u B.
Q7. Using properties of sets, show that.
(i) A∪ (A∩B) = A.
Ans: A ∪(A ∩ B) = A
We know that
A ⊂ A
A ∩ B ⊂ A
A ∪(A ∩ B) ⊂ A….. (1)
Also,
A ⊂ A ∪(A ∩ B)….. (2)
From (1) and (2),
A ∪(A ∩ B) = A
(ii) A∩ (A ∪ B)=A.
Ans: A ∩ (A U B) = A
Α∩ (A u B) = (A ∩ A) u (A ∩ B)
= A u (A ∩ B)
Α ∩ (Α υ Β) = A [from (1)]
Q8. Show that A ∩ B = A ∩ C need not imply B = C.
Ans: We can show this by taking some example sets.
Let, A = {1, 2, 3}
B = {2, 3, 4}
C = {2, 3, 5}
Then, A ∩ B = {2, 3} and A ∩ C = {2, 3} Thus, A ∩ B = A ∩ C even if B is not equal to C.
9. Let A and B be sets. If A ∩ X = B ∩ X = ⍉ and A ∪ X= B ∪ X for some set X, show that A = B.
(Hints A = A ∩(A ∪ X) , B = B ∩(B ∪ X) and use Distributive law)
Ans: A = A ∩ (A U X)
= A n (B U X)
= (A ∩ B) U (A ∩ X)
= (A ∩ B) U ⍉ (by distributive law)
A = A ∩ B eqn(1)
now B = B ∩ (B U X)
= B∩ (A U X)
= (B ∩ A) U (B ∩ X)
= (Β ∩ Α) U ⍉
B= (B ∩ A) eqn(2)
so by equal 1 & 2, A = B
hence proved
Q10. Find sets A, B and C such that A ∩ B, B ∩ C and A ∩ C are non-empty sets and A ∩ B ∩ C = ⍉.
Ans: Let A = {0, 1} , B = {1, 2} and C = {2, 0}
Accordingly,
A ∩ B = {1}
B ∩ C = {2}
A ∩ C = {0}
∴ A ∩ B, B ∩ C, and A ∩ C are non-empty
However, A ∩ B ∩ C = ⍉