Chapter 1.1: Sets and Their Representations
- Definition: A collection of well-defined objects.
- Elements: Individual objects in a set.
- Notation:
- Roster method: Listing elements.
- Set-builder form: Describing elements using a property.
- Types of Sets:
- Finite sets
- Infinite sets
- Empty set: Contains no elements.
- Singleton set: Contains only one element.
Chapter 1.2: Subsets
- Definition: A set contained within another set.
- Proper subset: A subset that is not equal to the original set.
- Subset notation: A ⊆ B means A is a subset of B.
Chapter 1.3: Equal Sets
- Definition: Sets that contain exactly the same elements.
- Equality notation: A = B means A and B are equal sets.
Chapter 1.5: Universal Set
- Definition: A set containing all elements under consideration in a particular context.
- Notation: U represents the universal set.
Chapter 1.6: Complement of a Set
- Definition: The set of all elements in the universal set that are not in a given set.
- Notation: A’ represents the complement of A.
Chapter 1.7: Union and Intersection of Sets
- Union: The set of elements that are in A or B or both.
- Intersection: The set of elements that are in both .
- Notation:
- A ∪ B represents the union
- A ∩ B represents the intersection
Key Concepts:
- Sets and their representations
- Subsets and proper subsets
- Equal sets
- Power sets
- Universal sets
- Complements
- Union and intersection of sets
Exercise 1.1
1. Which of the following are sets ? Justify your answer.
(i) The collection of all the months of a year beginning with the letter J.
(ii) The collection of ten most talented writers of India.
(iii) A team of eleven best-cricket batsmen of the world.
(iv) The collection of all boys in your class.
(v) The collection of all natural numbers less than 100.
(vi) A collection of novels written by the writer Munshi Prem Chand.
(vii) The collection of all even integers.
(viii) The collection of questions in this Chapter.
(ix) A collection of most dangerous animals of the world.
Ans :
Sets:
- (i) This is a set because the elements (months) are clearly defined and can be listed: January, June, July.
- (iv) This is a set because the elements (boys) are clearly defined and can be identified within the specific context of your class.
- (v) This is a set because the elements (natural numbers) are clearly defined and can be listed or described using a specific rule.
- (vi)This is a set because the elements (novels) are clearly defined and can be listed or identified based on the author.
- (vii) This is a set because the elements (even integers) are clearly defined and can be described using a specific rule.
- (viii) This is a set because the elements (questions) are clearly defined and can be listed or identified within the specific context of the chapter.
Not Sets:
- (ii) This is not a set because the term “most talented” is subjective and can lead to different interpretations, making the collection not well-defined.
- (iii) Similar to (ii), the term “best” is subjective and can vary based on different criteria, making the collection not well-defined.
- (ix) The term “most dangerous” is subjective and can depend on factors like habitat, behavior, and human interaction. This makes the collection not well-defined.
2. Let A = {1, 2, 3, 4, 5, 6}. Insert the appropriate symbol ∈ or ∉ in the blank spaces:
(i) 5. . .A (ii) 8 . . . A (iii) 0. . .A (iv) 4. . . A (v) 2. . .A (vi) 10. . .A
Ans :
(i) 5 ∈ A (5 is an element of A)
(ii) 8 ∉ A (8 is not an element of A)
(iii) 0 ∉ A (0 is not an element of A)
(iv) 4 ∈ A (4 is an element of A)
(v) 2 ∈ A (2 is an element of A)
(vi) 10 ∉ A (10 is not an element of A)
3. Write the following sets in roster form:
(i) A = {x : x is an integer and –3 ≤ x < 7}
(ii) B = {x : x is a natural number less than 6}
(iii) C = {x : x is a two-digit natural number such that the sum of its digits is 8}
(iv) D = {x : x is a prime number which is divisor of 60}
(v) E = The set of all letters in the word TRIGONOMETRY
(vi) F = The set of all letters in the word BETTER
Ans :
(i) A = {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {1, 2, 3, 4, 5}
(iii) C = {17, 26, 35, 44, 53, 62, 71, 80}
(iv) D = {2, 3, 5}
(v) E = {T, R, I, G, O, N, O, M, E, T, R, Y}
(vi) F = {B, E, T, T, E, R}
4. Write the following sets in the set-builder form :
(i) (3, 6, 9, 12} (ii) {2,4,8,16,32} (iii) {5, 25, 125, 625} (iv) {2, 4, 6, . . .}
(v) {1,4,9, . . .,100}
Ans :
(i) {x : x is a multiple of 3 and 1 ≤ x ≤ 12}
(ii) {x : x is a power of 2 and 2 ≤ x ≤ 32}
(iii) {x : x is a power of 5 and 5 ≤ x ≤ 625}
(iv) {x : x is an even natural number}
(v) {x : x is a square of a natural number and 1 ≤ x ≤ 100}
5. List all the elements of the following sets :
(i) A = {x : x is an odd natural number}
(ii) B = {x : x is an integer, 1 2 – < x < 9 2 }
(iii) C = {x : x is an integer, x 2 ≤ 4}
(iv) D = {x : x is a letter in the word “LOYAL”}
(v) E = {x : x is a month of a year not having 31 days}
(vi) F = {x : x is a consonant in the English alphabet which precedes k }.
Ans :
(i) A = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, …}
(ii) B = {-1, 0, 1, 2, 3, 4}
(iii) C = {-2, -1, 0, 1, 2}
(iv) D = {L, O, Y, A}
(v) E = {February, April, June, September, November}
(vi) F = {b, c, d, f, g, h, j}
6. 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}
(ii) {2, 3} (b) {x : x is an odd natural number less than 10}
(iii) {M,A,T,H,E,I,C,S} (c) {x : x is natural number and divisor of 6}
(iv) {1, 3, 5, 7, 9} (d) {x : x is a letter of the word MATHEMATICS}.
Ans :
(i) {1, 2, 3, 6} (c) {x : x is a natural number and divisor of 6}
(ii) {2, 3} (a) {x : x is a prime number and a divisor of 6}
(iii) {M,A,T,H,E,I,C,S} (d) {x : x is a letter of the word MATHEMATICS}
(iv) {1, 3, 5, 7, 9} (b) {x : x is an odd natural number less than 10}
Exercise 1.2
1. Which of the following are examples of the null set
(i) Set of odd natural numbers divisible by 20
(ii) Set of even prime numbers
(iii) { x : x is a natural numbers, x < 5 and x > 7 }
(iv) { y : y is a point common to any two parallel lines}
Ans :
(i) This set is empty because there are no odd numbers that are divisible by 2.
(ii) This set is also empty because 2 is the only even prime number, and it is not divisible by any other number except 1 and itself.
(iii) This set is empty because there are no natural numbers that are both less than 5 and greater than 7.
(iv) This set is empty because parallel lines never intersect, so there is no point that is common to both lines.
2. Which of the following sets are finite or infinite
(i) The set of months of a year
(ii) {1, 2, 3, . . .}
(iii) {1, 2, 3, . . .99, 100}
(iv) The set of positive integers greater than 100
(v) The set of prime numbers less than 99
Ans :
(i) This set has a fixed and limited number of elements (12 months).
Therefore, it is a finite set.
(ii) This set represents the set of all natural numbers, which continues indefinitely.
Therefore, it is an infinite set.
(iii) This set contains all natural numbers from 1 to 100, which is a limited number of elements.
Therefore, it is a finite set.
(iv) This set includes all positive integers starting from 101 and continuing indefinitely.
Therefore, it is an infinite set.
(v) While there might be a large number of prime numbers less than 99, it’s still a finite quantity.
You can eventually list all of them, even though it might take a long time.
Therefore, it is a finite set.
3. State whether each of the following set is finite or infinite:
(i) The set of lines which are parallel to the x-axis
(ii) The set of letters in the English alphabet
(iii) The set of numbers which are multiple of 5
(iv) The set of animals living on the earth
(v) The set of circles passing through the origin (0,0)
Ans :
(i) Infinite: There are an infinite number of lines parallel to the x-axis.
(ii) Finite: There are only 26 letters in the English alphabet.
(iii) Infinite: There are an infinite number of multiples of 5.
(iv) Finite: While the exact number is unknown, there is a finite number of animals living on Earth.
(v) Infinite: There are an infinite number of circles that can pass through the origin.
4. In the following, state whether A = B or not:
(i) A = { a, b, c, d } B = { d, c, b, a }
(ii) A = { 4, 8, 12, 16 } B = { 8, 4, 16, 18}
(iii) A = {2, 4, 6, 8, 10} B = { x : x is positive even integer and x ≤ 10}
(iv) A = { x : x is a multiple of 10}, B = { 10, 15, 20, 25, 30, . . . }
Ans :
(i) A = B Both sets contain the same elements.
(ii) A ≠ B The set B contains the element 18, which is not in set A.
(iii) A = B Both sets contain the same elements: all positive even integers less than or equal to 10.
(iv) A ≠ B The set B contains the element 15, which is not a multiple of 10. Therefore, A is not equal to B.
5. Are the following pair of sets equal ? Give reasons.
(i) A = {2, 3}, B = {x : x is solution of x 2 + 5x + 6 = 0}
(ii) A = { x : x is a letter in the word FOLLOW}
B = { y : y is a letter in the word WOLF}
Ans :
(i) A ≠ B
Let’s solve the quadratic equation x² + 5x + 6 = 0: (x + 2)(x + 3) = 0 x = -2 or x = -3
Therefore, B = {-2, -3}
Since A = {2, 3} and B = {-2, -3}, the sets A and B are not equal.
(ii) A = B
Both sets A and B contain the same elements: F, O, L, and W.
Therefore, A and B are equal sets.
6. 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 :
- B = D: Both sets contain the elements 1, 2, 3, and 4, even though the order is different.
- E = G: Both sets contain the elements -1 and 1.
Therefore, the pairs of equal sets are: B = D and E = G
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 }
(ii) { a, b, c } . . . { b, c, d }
(iii) {x : x is a student of Class XI of your school}. . .{x : x 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}
(v) {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 a triangle in the same plane}
(vii) {x : x is an even natural number} . . . {x : x is an integer}
Ans :
(i) { 2, 3, 4 } ⊂ { 1, 2, 3, 4,5 }
(ii) { a, b, c } ⊄ { b, c, d }
(iii) {x : x is a student of Class XI of your school} ⊂ {x : x 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}
(v) {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 a triangle in the same plane} (vii) {x : x is an even natural number} ⊂ {x : x is an integer}
2. Examine whether the following statements are true or false:
(i) { a, b } ⊄ { b, c, a }
(ii) { a, e } ⊂ { x : x is a vowel in the English alphabet}
(iii) { 1, 2, 3 } ⊂ { 1, 3, 5 }
(iv) { a } ⊂ { a, b, c }
(v) { 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 :
(i) False
(ii) True
(iii) False
(iv) True
(v) False
(vi) True.
3. Let A = { 1, 2, { 3, 4 }, 5 }. Which of the following statements are incorrect and why?
(i) {3, 4} ⊂ A (ii) {3, 4} ∈ A (iii) {{3, 4}} ⊂ A (iv) 1 ∈ A (v) 1 ⊂ A (vi) {1, 2, 5} ⊂ A (vii) {1, 2, 5} ∈ A (viii) {1, 2, 3} ⊂ A (ix) φ ∈ A (x) φ ⊂ A (xi) {φ} ⊂ A
Ans :
(i) {3, 4} ⊂ A
- This statement is incorrect.
- A proper subset contains all the elements of another set but is not equal to it.
- In this case, {3, 4} is not a proper subset of A because A contains the set {3, 4} as an element, not as a subset.
(ii) {3, 4} ∈ A
- This statement is correct.
- The symbol ∈ means “is an element of”.
- A contains the set {3, 4} as one of its elements.
(iii) {{3, 4}} ⊂ A
- This statement is correct.
- The set {{3, 4}} contains only one element: the set {3, 4}.
- Since A contains {3, 4} as an element, {{3, 4}} is a subset of A.
(iv) 1 ∈ A
- This statement is correct.
- 1 is one of the elements listed in A.
(v) 1 ⊂ A
- This statement is incorrect.
- 1 is an element, not a set. The symbol ⊂ can only be used between two sets.
(vi) {1, 2, 5} ⊂ A
- This statement is correct.
- The set {1, 2, 5} contains all the elements 1, 2, and 5, which are also elements of A.
(vii) {1, 2, 5} ∈ A
- This statement is incorrect.
- {1, 2, 5} is a set itself, not an element of A. A contains the individual elements 1, 2, and 5, but not the set {1, 2, 5}.
(viii) {1, 2, 3} ⊂ A
- This statement is incorrect.
- The element 3 is not contained within A as an individual element. It is part of the set {3, 4}, which is an element of A. Therefore, {1, 2, 3} is not a subset of A.
(ix) φ ∈ A
- This statement is incorrect.
- The empty set φ cannot be an element of another set.
(x) φ ⊂ A
- This statement is correct.
(xi) {φ} ⊂ A
- This statement is correct.
- The set {φ} contains only one element: the empty set φ. Since A contains the empty set as an element (even though it’s not explicitly listed), {φ} is a subset of A.
4. Write down all the subsets of the following sets
(i) {a} (ii) {a, b} (iii) {1, 2, 3} (iv) φ
Ans :
(i) Subsets of {a}:
- ∅ (empty set)
- {a}
(ii) Subsets of {a, b}:
- ∅ (empty set)
- {a}
- {b}
- {a, b}
(iii) Subsets of {1, 2, 3}:
- ∅ (empty set)
- {1}
- {2}
- {3}
- {1, 2}
- {2, 3}
- {1, 3}
- {1, 2, 3}
(iv) Subsets of φ:
- ∅ (the only subset of the empty set is itself)
5. Write the following as intervals :
(i) {x : x ∈ R, – 4 < x ≤ 6} (ii) {x : x ∈ R, – 12 < x < –10}
(iii) {x : x ∈ R, 0 ≤ x < 7} (iv) {x : x ∈ R, 3 ≤ x ≤ 4}
Ans :
(i) (-4, 6]
(ii) (-12, -10)
(iii) [0, 7)
(iv) [3, 4]
6. Write the following intervals in set-builder form :
(i) (– 3, 0) (ii) [6 , 12] (iii) (6, 12] (iv) [–23, 5)
Ans :
(i) (-3, 0) = {x ∈ R : -3 < x < 0}
(ii) [6, 12] = {x ∈ R : 6 ≤ x ≤ 12}
(iii) (6, 12] = {x ∈ R : 6 < x ≤ 12}
(iv) [-23, 5) = {x ∈ R : -23 ≤ x < 5}
7. What universal set(s) would you propose for each of the following :
(i) The set of right triangles. (ii) The set of isosceles triangles.
Ans :
Universal Set (U): A set that contains all elements under consideration in a particular context.
For the given sets of triangles, we can propose the following universal sets:
1. The set of right triangles:
- U: The set of all triangles.
This universal set would encompass all types of triangles, including right triangles, equilateral triangles, isosceles triangles, scalene triangles, etc.
2. The set of isosceles triangles:
- U: The set of all triangles.
Similar to the first case, the universal set for isosceles triangles could also be the set of all triangles. This would ensure that all possible isosceles triangles are included within the universal set.
8. 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 universal set (s) for all the three sets A, B and C
(i) {0, 1, 2, 3, 4, 5, 6}
(ii) φ
(iii) {0,1,2,3,4,5,6,7,8,9,10}
(iv) {1,2,3,4,5,6,7,8}
Ans :
A universal set for a collection of sets is a set that contains all the elements of those sets.
Therefore, the universal set for A, B, and C must contain all the elements of A, B, and C.
Let’s check each option:
- (i) {0, 1, 2, 3, 4, 5, 6}: This set contains all the elements of A, B, and C, so it could be a universal set.
- (ii) φ: The empty set cannot contain any elements, so it cannot be a universal set for any non-empty sets.
- (iii) {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}: This set contains all the elements of A, B, and C, and some additional elements, so it could also be a universal set.
- (iv) {1, 2, 3, 4, 5, 6, 7, 8}: This set contains all the elements of A, B, and C, so it could be a universal set.
The possible universal sets for A, B, and C are:
- {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
- {0, 1, 2, 3, 4, 5, 6, 7, 8}
- {1, 2, 3, 4, 5, 6}
Exercise 1.4
1. Find the union of each of the following pairs of sets :
(i) X = {1, 3, 5} Y = {1, 2, 3}
(ii) A = [ a, e, i, o, u} B = {a, b, c}
(iii) A = {x : x is a natural number and multiple of 3} B = {x : x is a natural number less than 6}
(iv) A = {x : x is a natural number and 1 < x ≤6 } B = {x : x is a natural number and 6 < x < 10 }
(v) A = {1, 2, 3}, B = φ
Ans :
(i) X ∪ Y = {1, 2, 3, 5}
(ii) A ∪ B = {a, b, c, e, i, o, u}
(iii) A ∪ B = {1, 2, 3, 4, 5, 6}
(iv) A ∪ B = {2, 3, 4, 5, 6, 7, 8, 9}
(v) A ∪ B = {1, 2, 3}
2. Let A = { a, b }, B = {a, b, c}. Is A ⊂ B ? What is A ∪ B ?
Ans :
Yes, A ⊂ B.
A ∪ B = {a, b, c}
3. If A and B are two sets such that A ⊂ B, then what is A ∪ B ?
Ans :
If A ⊂ B, then A ∪ B = B.
4. 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 ∪ B (ii) A ∪ C (iii) B ∪ C (iv) B ∪ D (v) A ∪ B ∪ C (vi) A ∪ B ∪ D (vii) B ∪ C ∪ D
Ans :
(i) A ∪ B = {1, 2, 3, 4, 5, 6}
(ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(iii) B ∪ C = {3, 4, 5, 6, 7, 8}
(iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}
(vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
(vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}
5. Find the intersection of each pair of sets of question 1 above
Ans :
(i) A = {1, 3, 5} Y = {1, 2, 3} A ∩ Y = {1, 3}
(ii) 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} 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 } A ∩ B = ∅ (empty set)
(v) A = {1, 2, 3}, B = φ A ∩ B = φ (empty set)
6.If A = { 3, 5, 7, 9, 11 }, B = {7, 9, 11, 13}, C = {11, 13, 15}and D = {15, 17}; find
(i) A ∩ B (ii) B ∩ C (iii) A ∩ C ∩ D (iv) A ∩ C (v) B ∩ D (vi) A ∩ (B ∪ C)
(vii) A ∩ D (viii) A ∩ (B ∪ D) (ix) ( A ∩ B ) ∩ ( B ∪ C ) (x) ( A ∪ D) ∩ ( B ∪ C)
Ans :
(i) A ∩ B = {7, 9, 11}
(ii) B ∩ C = {11, 13}
(iii) A ∩ C ∩ D = ∅ (empty set, as no element is common to all three sets)
(iv) A ∩ C = {11}
(v) B ∩ D = ∅ (empty set)
(vi) A ∩ (B ∪ C) = {7, 9, 11} (first find B ∪ C, then intersect with A)
(vii) A ∩ D = ∅ (empty set)
(viii) A ∩ (B ∪ D) = {7, 9, 11} (first find B ∪ D, then intersect with A)
(ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} (first find A ∩ B and B ∪ C, then intersect the results) (x) (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17} ∩ {7, 9, 11, 13, 15} = {7, 9, 11, 15}
7. If A = {x : x is a natural number }, B = {x : x is an even natural number} C = {x : x is an odd natural number}andD = {x : x is a prime number }, find
(i) A ∩ B (ii) A ∩ C (iii) A ∩ D (iv) B ∩ C (v) B ∩ D (vi) C ∩ D
Ans :
(i) A ∩ B = {x : x is an even natural number} = B
(ii) A ∩ C = {x : x is an odd natural number} = C
(iii) A ∩ D = {x : x is a prime number} = D
(iv) B ∩ C = ∅ (empty set, as even and odd numbers have no common elements)
(v) B ∩ D = {2} (2 is the only even prime number)
(vi) C ∩ D = {x : x is an odd prime number} = D – {2} (all prime numbers except 2)
8. 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 }
(ii) { a, e, i, o, u } and { c, d, e, f }
(iii) {x : x is an even integer } and {x : x is an odd integer}
Ans :
(i) {1, 2, 3, 4} and {x : x is a natural number and 4 ≤ x ≤ 6 }
- The second set contains elements 4, 5, and 6.
- They share the element 4.
- Therefore, they are not disjoint.
(ii) { a, e, i, o, u } and { c, d, e, f }
- They share the element ‘e’.
- Therefore, they are not disjoint.
(iii) {x : x is an even integer } and {x : x is an odd integer}
- Even and odd integers have no common elements.
- Therefore, they are disjoint.
9. 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 (ii) A – C (iii) A – D (iv) B – A (v) C – A (vi) D – A (vii) B – C
(viii) B – D (ix) C – B (x) D – B (xi) C – D (xii) D – C
Ans :
10. If X= { a, b, c, d } and Y = { f, b, d, g}, find (i) X – Y (ii) Y – X (iii) X ∩ Y
Ans :
1. X – Y
- This represents the set of elements in X that are not in Y.
- X – Y = {a, c}
2. Y – X
- This represents the set of elements in Y that are not in X.
- Y – X = {f, g}
3. X ∩ Y
- This represents the intersection of X and Y, or the set of elements that are in both X and Y.
- X ∩ Y = {b, d}
11. If R is the set of real numbers and Q is the set of rational numbers, then what is
R – Q?
Ans :
R – Q represents the set of real numbers that are not rational numbers. This is the set of irrational numbers.
Therefore, R – Q = {x ∈ R : x is irrational}.
12. 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.
(ii) { a, e, i, o, u } and { a, b, c, d }are disjoint sets.
(iii) { 2, 6, 10, 14 } and { 3, 7, 11, 15} are disjoint sets.
(iv) { 2, 6, 10 } and { 3, 7, 11} are disjoint sets.
Ans :
Exercise 1.5
1. 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′ (ii) B′ (iii) (A ∪ C)′ (iv) (A ∪ B)′ (v) (A′)′ (vi) (B – C)′
Ans :
1. A’
- This represents the complement of A, or the set of elements in U that are not in A.
- A’ = {5, 6, 7, 8, 9}
2. B’
- This represents the complement of B.
- B’ = {1, 3, 5, 7, 9}
3. (A ∪ C)’
- First, find A ∪ C: A ∪ C = {1, 2, 3, 4, 5, 6}
- Then, find the complement of A ∪ C.
- (A ∪ C)’ = {7, 8, 9}
4. (A ∪ B)’
- First, find A ∪ B: A ∪ B = {1, 2, 3, 4, 6, 8}
- (A ∪ B)’ = {5, 7, 9}
5. (A’)’
- This represents the complement of the complement of A, which is equivalent to A itself.
- (A’)’ = A = {1, 2, 3, 4}
6. (B – C)’
- First, find B – C: B – C = {2, 8} (elements in B but not in C)
- Then, find the complement of B – C.
- (B – C)’ = {1, 3, 4, 5, 6, 7, 9}
2. If U = { a, b, c, d, e, f, g, h}, find the complements of the following sets : (i) A = {a, b, c} (ii) B = {d, e, f, g} (iii) C = {a, c, e, g} (iv) D = { f, g, h, a}
Ans :
1. A’
- This represents the complement of A, or the set of elements in U that are not in A.
- A’ = {d, e, f, g, h}
2. B’
- This represents the complement of B.
- B’ = {a, b, c, h}
3. C’
- This represents the complement of C.
- C’ = {b, d, f, h}
4. D’
- This represents the complement of D.
- D’ = {b, c, d, e}
3. 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}
(ii) { x : x is an odd natural number }
(iii) {x : x is a positive multiple of 3}
(iv) { x : x is a prime number }
(v) {x : x is a natural number divisible by 3 and 5}
(vi) { x : x is a perfect square }
(vii) { x : x is a perfect cube}
(viii) { x : x + 5 = 8 }
(ix) { x : 2x + 5 = 9}
(x) { x : x ≥ 7 }
(xi) { x : x ∈ N and 2x + 1 > 10 }
Ans :
4. 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 ∪ B)′ = A′ ∩ B′
Ans :
De Morgan’s Laws state that for any sets A and B:
- (A ∪ B)’ = A’ ∩ B’
- (A ∩ B)’ = A’ ∪ B’
Given:
- U
= {1, 2, 3, 4, 5, 6, 7, 8, 9}
- A = {2, 4, 6, 8}
- B = {2, 3, 5, 7}
Let’s verify the first law: (A ∪ B)’ = A’ ∩ B’
Step 1: Find A ∪ B
- A ∪ B
- = {1, 2, 3, 4, 5, 6, 7, 8}
Step 2: Find (A ∪ B)’
- (A ∪ B)’ = {9}
Step 3: Find A’ and B’
- A’ = {1, 3, 5, 7, 9}
- B’ = {1, 4, 6, 8, 9}
Step 4: Find A’ ∩ B’
- A’ ∩ B’ = {9}
Conclusion: Since (A ∪ B)’ = {9} and A’ ∩ B’ = {9}, we have verified that (A ∪ B)’ = A’ ∩ B’ is true for the given sets.
5. Draw appropriate Venn diagram for each of the following : (i) (A ∪ B)′, (ii) A′ ∩ B′, (iii) (A ∩ B)′, (iv) A′ ∪ B′
Ans :
6. 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.
Here’s the explanation:
- A is defined as the set of all triangles with at least one angle different from 60°.
- A’ is the complement of A, meaning it contains all the elements in the universal set U (all triangles) that are not in A.
- If a triangle has at least one angle different from 60°, it cannot be an equilateral triangle (since equilateral triangles have all angles equal to 60°).
- Therefore, A’ must contain all the triangles that have all angles equal to 60°.
- A’ is the set of all equilateral triangles.
7. Fill in the blanks to make each of the following a true statement :
(i) A ∪ A′ = . . . (ii) φ′ ∩ A = . . . (iii) A ∩ A′ = . . . (iv) U′ ∩ A = . . .
Ans :
(i) A ∪ A′ = U.
(ii) φ′ ∩ A = A
- The complement of the empty set is the universal set (U).
- So, U ∩ A = A.
(iii) A ∩ A′ = φ
(iv) U′ ∩ A = φ
- The complement of the universal set is the empty set (φ).
- So, φ ∩ A = φ.
