SETS CHAPTER 1
No object should be repeated in the collection; it means that the objects are distinct, e.g., the set of letters of the word SCHOOL is {S, C, H, O, L}.
Objects in the set are called elements or members of the set. Sets are denoted by capital letters; elements are denoted by small letters. If x is an element of set A, then we say x ∈ A, read as x belongs to A
Consider the following collections of objects.
(i) The collection of all capitals of states of India is a set.
(ii) The collection of beautiful girls in a class is not a set.
(iii) Collection of prime factors of 20 is a set.
In the above collections, (i) and (iii) are well-defined collections, and (ii) is not a well-defined collection.
We will have frequent interaction with some sets. So, we reserve some letters to represent those sets, as shown below.
N : Set of natural numbers
W : Set of whole numbers
Z : Set of integers
Q : Set of rational numbers
R : Set of real numbers
C : Set of complex numbers
1. Which of the following collections are sets? Justify your answer.
(i) A collection of the most dangerous animals in this world
(ii) A collection of names of months that start with J
Sol: We know that a set is a well-defined collection of distinct objects.
The collection of most dangerous animals in this world is not well defined. So, this collection is not a set.
The collection of names of months that start with ‘J’ is January, June, and July. So, this collection is a set.
Try yourself:
1. If a set A = { a , e , i , o , u }, then insert appropriate symbol in each of the following blank spaces.
(i) a___A
(ii) 3___A Ans: (i) ∈ (ii) ∉
1.1.2 Representation of a Set
A set can be described in two ways: roster form and set builder form.
Roster Form: Roster form, also known as tabular form or enumeration, is a way of representing a set by explicitly listing all its elements inside curly braces. The elements are separated by commas.
The order in which the elements are listed does not matter, and each element is typically listed only once, as sets do not contain duplicate elements, e.g., the set of all even numbers less than 10 can be represented as {2, 4, 6, 8}.
Roster form is a simple and straightforward way to represent sets when the number of elements is small and manageable. However, for larger sets or sets with a specific pattern
or condition, set-builder notation may be more practical.
Set Builder Form: The set - builder notation is a way to describe a set by specifying a rule or condition that its members must satisfy. It is a concise and convenient way to represent sets without explicitly listing all the elements.
The general form of set builder notation is { x : P ( x )}. Here, x represents the variable that takes on values in the set, and P(x) is a predicate (a condition or rule) that determines whether an element belongs to the set, e.g., the set of all natural numbers less than 10 can be represented as {x / 0 < x < 10, x ∈ N}; this is called set builder form.
2. List elements of the following set in roster form.
19 :is an integer, 22 Cxxx
Sol: Given that the set
19 :is an integer, 22 Cxxx
Here – 0.5 < x < 4.5
It means that x takes the values 0, 1, 2, 3, 4.
∴ the set C is {0, 1, 2, 3, 4}
Try yourself:
2. Write the set builder form of the set D = {0, 3, 6, 9, 12, ......}.
Ans: D = { x : x is a no n- negative multiple of 3}
TEST YOURSELF
1. The group of tallest boys is (1) a null set (2) a finite set (3) an infinite set (4) not a set
2. Which of t he following is the correct set builder form for the set {–3, –2, –1, 0, 1, 2, 3}?
(1) {x : x is a whole number and –3 ≤ x ≤ 3}
(2) {x : x is an irrational number and –3 ≤ x ≤ 3}
(3) {x : x is an integer and –3 ≤ x ≤ 4}
(4) {x : x is an integer and –3 ≤ x ≤ 3}
3. Which of the following is the correct set builder form for the set {4, 5, 6, 7, 8}?
(1) {x : x ∈ N, 3 < x ≤ 9}
(2) {x : x ∈ N, 3 ≤ x < 9}
(3) { x : x ∈ N, 3 < x < 9}
(4) {x : x ∈ N, 3 ≤ x ≤ 9}
4. Which of the following is the correct set builder form for the set {1, 3, 9, 27, 81, 243}?
(1) {x : x = 3n, n ∈ Z, 0 ≤ n < 6}
(2) {x : x = 3n, n ∈ Z, 0 ≤ n < 5}
(3) {x : x = 3n, n ∈ Z, 0 ≤ n < 5}
(4) {x : x = nn, n ∈ Z, 0 ≤ n < 4}
5. Which of the following is the correct roster form for the set of all letters in the word ‘REPRESENTATION’?
(1) {R, E, P, R, E, S, E, N, T, A, T, I, O, N}
(2) {R, E, P, E, S, N, T, A, I, O, N}
(3) {R, E, P, S, N, T, A, I, O, N}
(4) {R, E, P, S, N, T, A, I, O}
6. Which of the following is the correct roster form for { x ∈ N : x is a prime number, 1 < x < 30}?
(1) {3, 5, 7, 11, 13, 17, 19, 23, 29}
(2) {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}
(3) {2, 3, 5, 7, 11, 13, 17, 19, 23}
(4) {1, 3, 5, 7, 11, 13, 17, 19, 23}
Answer Key
(1) 4 (2) 4 (3) 3 (4) 1 (5) 4 (6) 2
1.2 TYPES OF SETS
Variou s types of sets exist, based on the number of elements in the set.
Let A be a set. The number of elements in the set A is called the cardinal number of set, and it is denoted by n(A).
Example: If A = {a,e,i,o,u}, then n(A) = 5.
1.2.1 Empty Set
A set is said to be an empty or void set if it has no element. It is denoted by φ (phi).
n( φ ) = 0.
If a set has at least one element, then that set is called a non-empty set.
Example: Let C = {x : x is a even prime number greater than 2}
3. Which of the following are empty sets?
(i) The set of all real numbers whose square is –1
(ii) A = {x : x ∈ N, 1 < x < 2}
(iii) The set of all vowels in the word ‘RYTHMS’
Sol: (i) There is no real number whose square i s negative so that the set of all the real numbers whose square is –1 is empty set.
(ii) There is no natural number between two successive natural numbers. So that A = {x : x ∈ N, 1 < x < 2} is empty set.
(iii) There is no vowel in the word rytham. So that A = {x : x ∈ N, 1 < x < 2} is empty set. The set of all vowels in the word ‘RYTHMS’ is empty set.
Try yourself:
3. Which of the following are non-empty sets?
(i) The set of all common points of two perpendicular lines in two-dimensional geometry
(ii) The set of all prime numbers which are divisible by 2
Ans: (i), (ii)
1.2.2
Singleton Set
A set is said to be a singleton set if it has only one element. Set S is a singleton set if and only if n(S) = 1.
Example: Let C = { x : x is an even prime number} = {2}
4. Check whether the set {x : x ∈ N, x2 = 9} is a singleton set.
Sol: Given, x ∈ N, x2 = 9; it implies that x = 3 only. Therefore, the given set in roster form is {3}, which is a singleton set.
Try yourself:
4. Check whether the set { x : x is a root of x2 – 5x + 6 = 0} is a singleton set.
Ans: No
1.2.3
Finite Set
A set A is called a finite set if it is either an empty set or the elements can be counted by natural numbers and the counting terminates at a certain natural number, say n
The set F is a finite set if and only if n(F) = n, where n is either zero or a particular natural number.
Example: Set of all vowels in English alphabet is a finite set.
5. Check whether the set of prime numbers less than 25 is a finite set.
Sol: The set of all prime numbers less than 25 is A = {2, 3, 5, 7, 11, 13, 17, 19, 23}.
Here, number of elements in the set A is 9; so, set A is a finite set.
Try yourself:
5. Check whether the set of vowels in English is finite.
Ans: Yes, it is finite.
1.2.4 Infinite Set
A set A is called an infinite set if it is not a finite set. The cardinal number of infinite sets is not defined.
Example: Set of all natural numbers is an infinite set.
6. Let {} 2 :,is even SxxZx =∈ ; show that the set S is an infinite set.
Sol: The given set is {} 2 :,is even SxxZx =∈ .
In the roster form, it is {....,–6, –4, –2, 0, 2, 4, 6, .....}, and it is an infinite set.
Try yourself:
6. Check whether the set set of all integers which are multiples of 3 is an infinite set.
Ans: Yes
1.2.5 Intervals
An interval is a subset of real numbers that contains all real numbers lying between two specified numbers.
To represent a particular set of real numbers, we use the interval notation. There are different intervals, as shown below.
1. Open interval ( a,b ) is a set of all real numbers ‘x’, such that a < x < b, i.e., {x : x ∈ R,a < x < b}
2. Closed interval [ a,b ] is a set of all real numbers ‘x’, such that a ≤ x ≤ b, i.e., {x : x ∈ R,a ≤ x ≤ b}
3. Semi-closed interval [ a,b ) is a set of all real numbers ‘x’, such that a ≤ x < b, i.e., {x : x ∈ R,a ≤ x < b}
4. Semi -closed interval ( a,b ] is a set of all real numbers ‘x’, such that a < x ≤ b,
i.e., {x : x ∈ R,a < x ≤ b}
5. The set of all real numbers can be represented as (– ∞ , ∞).
All these intervals are infinite sets.
7. Write the following subsets of R as intervals.
(i) {} :,46xxRx∈−<≤
(ii) {} :,10xxRx∈≤
Sol: (i) {} :,46xxRx∈−<≤
Since, in the set, –4 is not included and 6 is included, the interval is a semi-closed interval. So, the interval notation for the set
{x : x ∈ R, –4 < x ≤ –6} is (–4, 6].
(ii) {} :,10xxRx∈≤
Since, in the above set, 10 is included and all the elements of the above set are less than or equal to 10, the interval notation of the set { x : x ∈ R, x ≤ 10} is (–∞, 10].
Try yourself:
7. Write the set builder form of the interval [–20, 3).
Ans: {x : x ∈ R, –20 ≤ x < 3}
1.2.6 Equivalent Sets
Two finite sets A and B are said to be equivalent sets if and only if their cardinal number is same; which means n(A) = n(B).
8. If A = {x : x∈N, x is a perfect square less than 30} and B is the set of all vowels in the English alphabet, then show that the sets A and B are equivalent sets.
Sol: The set A is {1, 4, 9, 16, 25} and the set B is {a, e, i, o, u}.
Here, n(A) = n(B) = 5
Hence, both the sets are equivalent sets.
Try yourself:
8. Check whether the sets ‘roots of the equations x2 – 5x + 6 = 0’ and ‘roots of the quadratic equation x2 – 1 = 0, are equivalent sets.
Ans: Yes
1.2.7 Equal Sets
Two finite sets A and B are said to be equal sets if and only if each element of set A is an element of set B and if each element of set B is an element of set A.
If sets A and B are equal then we write it as A=B. We write A ≠ B when AandB are not equal.
All equal sets are equivalent sets, but all equivalent sets may not be equal sets. A = {1, 2, 3, 4, 5} and B = { a,e,i,o,u } are equivalent sets but not equal sets.
9. Show that the set of letters needed to spell ‘CATARACT’ and the set of letters needed to spell ‘TRACT’ are equal.
Sol: The set of letters needed to spell ‘CATARACT’ is {A,C,T,R} and the set of letters needed to spell ‘TRACT’ is {A,C,T,R}.
Observing the above two sets, both sets have same elements. So, the two sets are equal sets.
Try yourself:
9. Show that the set of vowels in the English alphabet and the set of all single-digit odd numbers are not equal sets but equivalent sets
TEST YOURSELF
1. Which of the following is a null set?
(1) {0}
(2) {x : x ∈ R,x > 0 or x < 0}
(3) {x : x ∈ R,x2 = 4 or x = 3}
(4) {x : x ∈ R,x2 + 1 = 0, x ∈ R}
2. The set A = { x : x ∈ R , x 2 =16, and 2x = 6} equals to (1) φ (2) {14, 3, 4}
(3) {3} (4) {4}
3. In set-builder method, the null set is represented by
(1) { } (2) φ
(3) {x:x≠x} (4) {x:x=x}
4. Among the following, a singleton set is
(1) {x : |x| < 1, x ∈ z}
(2) {x : |x| = 5, x ∈ z}
(3) {x:x2 = 1, x ∈ z}
(4) {x:x2 + x + 1 = 0, x ∈ R}
5. The set of all even natural numbers is a/an
(1) infinite set
(2) finite set
(3) null set
(4) singleton set
Answer Key
(1) 4 (2) 1 (3) 3 (4) 1
(5) 1
1.3 COMPARABILITY OF SETS
Two sets A and B are comparable if and only if either both the sets are equal or each element of one set is also an element of the other set. Otherwise, the sets A and B are said to be incomparable sets.
1.3.1 Subset
Let AandB be any two non-empty sets. If every element of A is an element of B, then A is called a subset of B, and it is denoted as A⊂ B, which is read as ‘ A is subset or equal to B.’
1. An empty set is a subset of any set.
2. If there is at least one element of A which is not an element of B , then the set A is not a subset of B , and it is denoted as A ⊄ B.
Proper subset: If A is a non-empty subset of B and A ≠ B, then A is said to be a proper subset of B, and it is denoted as A ⊂ B.
Example: The set of all natural numbers is a proper subset of the set of all real numbers.
Improper Subset: Improper subset of set A is a subset of A containing all the elements of the set A.
Every set is an improper subset of itself. The total number of subsets of a finite set containing n elements is 2n; number of proper subsets is 2n – 1.
Properties of subset:
1. Let A,B,C be three sets. If A ⊂ B and B ⊂ C, then A ⊂ C
2. If A ⊂ B and B ⊂ A, then A = B.
10. 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. Then, find the value of m + n
Sol: The number of subsets of the set having m elements is 2m .
Similarly, 2n is the number of subsets of the set having n elements.
Given that 2m – 2n = 56.
The powers of 2 are 2, 4, 8, 16, 32, 64, 128,......
The difference of 64 and 8 is 56.
Hence, 2m = 64 and 2n = 8.
Therefore, m + n = 6 + 3 = 9.
Try yourself:
10. What is the number of subsets of the set {a, b, c}? Write all the subsets.
Ans: Number of subsets = 8 and they are
1.3.2 Super Set
A set A is called a super set of a set B if the set B is a subset of A. If A is super set of B, then this can be written as A ⊇ B. It is read as A is super set of B.
Set of real numbers is super set of all natural numbers.
Set of all letters of the English alphabet is the super set of all vowels.
11. If X = {x|x is a neutral number} and
Y : { x | x is an integer}, then prove that Y is super set of the set X
Sol: Given, X = {x|x is a neutral number}
X = {1, 2, 3, 4,…}
Y = {…–4, –3, –2, –1, 0, 1, 2, 3…}
Hence, the set Y is a super set of X
Try yourself:
11. Among the sets A = {x : x is a root of x2 – 8x + 12 = 0} and B = {2, 4, 6}, which set is the super set of the other set?
Ans: B ⊇ A
1.3.3 Power Set
The set of all subsets of a non-empty set A is called a power set of A, and it is denoted by P(A).
i. If A is a finite set with n elements, then the number of elements of power set of A is given by n(P(A)) = 2n
ii. P(A) = P(B) ⇔ A = B
iii. P(A) = {S : S ⊆ A}; the minimum number of elements of a power set of a non-empty set is 2.
iv. P(A) ∩ P(B) = P(A ∩ B)
v. P(A) ∪ P(B) ⊂ P(A ∪ B)
vi. The number of elements in P ( φ ) is one, and it is P( φ ) = { φ }.
12. Let n ( A ) = p , n ( B ) = q and the number of elements in the power sets of A be 60 more than the number of elements in the power set of B. Then, find the value of p + q.
Sol: 2p – 2q = 60 = 64 – 4
2p – 2q = 26 – 22 It implies that p = 6, q = 2, so that p + q = 8.
Try yourself:
12. What is the power set of the set { φ }?
Ans: { φ , { φ }}
1.3.4 Universal Set
A set is said to be a universal set if it is a super set of any set. A set containing all the elements of all sets in a given context is called a universal set, and it is denoted by U.
13. What is the universal set in the set of intervals?
Sol: Since all the intervals are subsets of a set of real numbers, the set of all real numbers is a universal set in this context.
Try yourself:
13. What is the universal set when we study two-dimensional coordinate geometry?
Ans: Set of all points in xy–plane
TEST YOURSELF
1. Let A={1,{1},2,{3,4},ϕ,{ϕ}}; Consider the following statements (where ϕ is null set).
(i) 1 ∈ A (ii) {1} ⊂ A
(iii) {1} ∈ A (iv) {2} ∈ A
(v) φ⊂ A (vi) {ϕ} ⊂ A
(vii) { φ } ∈ A (viii) {1,{3}} ∈ A
Ho w many of the above statements are correct?
(1) 5 (2) 6
(3) 7 (4) 8
2. The number of non-empty proper subsets of {x/x < 5, x ∈ N} is
(1) 16 (2) 15
(3) 13 (4) 14
3. If A = {1, 3, 5, B} and B ={2, 4}, then (1) 4 ∈ A (2) {4} ⊂ A
(3) B ⊂ A (4) B ∈ A
4. The number of elements in the power set P(S) of the set S = { {ϕ}, 1, {2, 3} } is (1) 16 (2) 8 (3) 9 (4) 3
5. If A = {ϕ, {ϕ}}, then the power set of A is (1) A
(2) {ϕ, { ϕ }, A}
(3) {ϕ, { ϕ }, {{ϕ}}, A} (4) {ϕ, { ϕ }}
Answer Key
(1) 2 (2) 2 (3) 4 (4) 2 (5) 3
1.4 OPERATIONS ON SETS
Sets, fundamental in mathematics, represent collections of distinct elements. Operations on sets include union (merging sets), intersection (common elements), difference (unique elements in one set), and complement (elements not in a set). These operations form the basis for various mathematical and logical analyses in diverse fields.
1.4.1
Venn Diagrams
Venn diagrams are visual representations used to illustrate relationships between different sets. Circular or overlapping shapes depict commonalities and differences among sets. Each circle represents a set, and overlapping areas show shared elements. Rectangle represents universal set in this context. Venn diagrams are widely employed in mathematics, logic, statistics, and various other fields for clear conceptual visualisation.
1.4.2
Union of Sets
Let A and B be any two sets. The union of two sets is a set having elements either in the set A or in the set B, or in both the sets. The union of two sets AandB is represented as A ∪ B.
(i) Mathematically, A∪B can be written as A ∪ B= {x : x ∈ A or x ∈ B}.
(ii) If n(A) = p and n(B) = q, then max. of
{p, q} ≤ n(A ∪ B) ≤ p+q
(iii) The shaded region in the following Venn diagram represents A ∪ B B A
14. Let U be the set of all single-digit natural numbers, A be the set of all single-digit even natural numbers, B be the set of all singledigit odd natural numbers. Find A ∪ B
Sol: Given: A = {2, 4, 6, 8} and B = {1, 3, 5, 7, 9}. A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Try yourself:
14. Let A = {x : x ∈ Z, –3 ≤ x ≤ 3} and B = {x : x ∈ N, –4 ≤ x ≤ 2}. Then, find A ∪ B. Ans: {–3, –2, –1, 0, 1, 2}
1.4.3 Intersection of Sets
Let A and B be any two sets. The intersection of two sets is the set of all those elements which are common to both A and B. The intersection of two sets AandB is represented as A ∩ B
(i) Mathematically, A ∩ B can be written as A ∩ B= {x : x ∈ A and x ∈ B}.
(ii) If n(A) = p and n(B) = q, then 0 ≤ n(A ∩ B) ≤ min. of {p, q}
(iii) The shaded region in the following Venn diagram represents A ∩ B. U
Disjoint sets: Two or more sets are said to be disjoint sets if the intersection of those sets is an empty set. Venn diagram for two disjoint sets is as shown below.
15. Let A be the set of all single-digit even natural numbers and B be the set of all single-digit odd natural numbers. Show that these two sets are disjoint sets.
Sol: Given: A = {2, 4, 6, 8} and B = {1, 3, 5, 7, 9}. Then, A ∩ B = { } is an empty set. So, A and B are disjoint sets.
Try yourself:
15. Let A = {x : x∈Z, –3 ≤ x ≤ 3} and B = {x : x ∈N, –4 ≤ x ≤ 2}. Then, find A ∩ B Ans: {3}
1.4.4 Complement of a Set
Let U be the universal set, and let A be any set such that A⊆U. Then, the complement of the set A with respect to U is denoted by A', and it is defined as A'=U–A. It is the set of all elements which do not belong to A.
A' = {x : x ∈ U and x ∉A }
The shaded region in the Venn diagram represents the complement of the set, as shown below.
i. A ∪ A' = U
ii. A ∩ A ' = φ
iii. If A ⊆ B, then B' ⊆ A'
iv. φ ' = U
v. U ' = φ
vi. (A')' = A
16. Let the universal set in this context be the set of all natural numbers. Let A be the set of all even natural numbers and B be the set of all odd numbers. Then, find A'.
Sol: The complement of the set A is U – A Hence, U–A is the set of all odd numbers.
∴ , A' = {1, 3, 5,......}
Try yourself:
16. If U = {1, 2, 3, 4, 5, 6, 7, 8, 9} and A = {2, 4, 6, 7, 9}, then find (A')'
Ans: {2, 4, 6, 7, 9} = A
1.4.5 Difference of Sets
1. Let A and B be two sets. The difference of two sets A and B is either A–B or B–A.
2. The set A–B is the set of all elements of the set A which do not belong to B; i.e., A–B = {x :x ∈ A and x ∉ B}
3. A–B = A∩B' and n(A–B) = n(A) – n(A∩B)
4. If n ( A ) = p and n ( B ) = q , then max. of {p–q, 0} ≤ n(A - B) ≤ p
5. The shaded region of the Venn diagram shown below represents the set A–B. B
U
Properties of complementary sets
6. The set B–A is the set of all elements of the set B which do not belongs to A
(i) B–A = {x : x ∈ B and x ∉ A}
(ii) B–A = A' ∩ B
7. If n(A) = p and n(B) = q, then max. of {q–p, 0} ≤ n(B - A) ≤ q.
8. The shaded region in the Venn diagram shown below represents the set B–A
1.4.6 Symmetric Difference of Sets
The symmetric difference of two sets A and B is denoted as A ∆ B and it is defined as A ∆ B = (A–B) ∪ (B–A), i.e., A ∆ B = {x : x ∈ A, x ∉ B or x ∈ B, x ∉ A}
The shaded region in the Venn diagram represents complement of a set. B
A B U
9. For any two sets A and B the sets A–B, A ∩ B,B–A are pair-wise disjoint sets.
10. Properties of difference of sets
i. If A–B=A , then A ∩ B = φ and vice versa.
ii. (A–B) ∩ B = φ
iii. (A–B) ∪ B = A ∪ B
iv. (A–B) ∪ (B–A) = (A ∪ B)–(A ∩ B)
v. A – (B ∩ C) = (A–B) ∪ (A–C)
vi. A – (B ∪ C) = (A–B) ∩ (A–C)
vii. A ∩ (B–C) = (A ∩ B) – (A ∩ C)
17. If U = {3, 4, 5,.... 20, 21}, A = {3, 6, 9, 12, 15, 18, 21}, and B = {4, 6, 8, 12, 20}, then show that A–B = A ∩ B'
Sol: From the given sets:
A–B = {3, 9, 15, 18, 21} and B'=U–B
B' = {3, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21}
Consider A ∩ B' = {3, 9, 15, 18, 21}
Therefore, A–B=A ∩ B'.
Try yourself:
17. If U = {3, 4, 5,....,20, 21}, A = {3, 6, 9, 12, 15, 18, 21}, and B = {4, 6, 8, 12, 20}, then show that B–A'=A' ∩ B.
18. If {} /,03AxxRx=∈<< and {} /,15BxxRx=∈≤≤ , then find A ∆ B.
Sol: Given: A = (0,3) and B = [1, 5].
We know that A ∆ B= (A–B) ∪ (B–A).
Hence, A–B = (0,1) and B–A = [3, 5].
Therefore, A ∆ B= (0, 1) ∪ [3, 5].
Try yourself:
18. Find the symmetric difference of two sets P = {a,b,c,d,e,f,g,h } and Q = {c,f,g,k}. Ans: {ke,d,b,a,}
TEST YOURSELF
1. If A , B , and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C, then (1) A=C (2) B=C (3) A ∩ B = φ (4) A=B ∪ C
2. Let A = { x : x is a multiple of 3} and B = {x/x is a multiple of 5}. Then, A ∩ B is given by
(1) {3, 6, 9, .....}
(2) {5, 10, 15, 20, ....}
(3) {15, 30, 45, ....}
(4) {30, 60, 90, ......}
3. If aN = {ax : x ∈ N}, then the set 3N ∩ 7N =
(1) {3, 6, 9, 12, ..}
(2) {7, 14, 21, 28, ....} (3) {21, 42, 63, 84, ...} (4) {5, 10, 15, ......}
4. If X = { x | x is a prime number} and Y = {x|x is a natural number}, then X∪Y is equal to (1) X (2) Y (3) N (4) X–Y
5. The symmetric difference of A = {1, 2, 3} and B = {3, 4, 5} is (1) {1, 2} (2) {1, 2, 4, 5} (3) {4, 3} (4) {2, 5, 1, 4, 3}
6. For two sets, A ∪ B=A if (1) B ⊂ A (2) A ⊂ B (3) A ≠ B (4) A=B
7. If A = { x:x is a multiple of 3} and, B={x:x is a multiple of 5}, then A–B is
(1) A ∩ B (2) AB ∩
(3) AB ∩ (4) AB ∩
Answer Key
(1) 2 (2) 3 (3) 3 (4) 2
(5) 2 (6) 1 (7) 2
1.5 LAWS OF ALGEBRA OF SETS
The laws of algebra of sets are a set of fundamental principles that govern the manipulation and relationships among sets. Sets are a fundamental concept in mathematics and are used to group together distinct elements. The algebra of sets provides a systematic way to perform operations on sets, similar to the algebraic operations performed on numbers. Some key laws of algebra of sets are explained below.
1.5.1 Idempotent Law
For any set A, we have
(i) A ∪ A=A
(ii) A ∩ A=A
19. If A = {1,2,3}, then show that A ∪ A = A
Sol: A ∪ A= {1,2,3} ∪{1,2,3}={1,2,3}= A
Try yourself:
19. If A = {1, 2, 3}, then show that A ∩ A = A
1.5.2 Identity Law
1. For any set A, we have
i. A ∪φ =A ii. A ∩ U=A, where φ is an empty set and U is a universal set, in this context.
2. The set φ is the identity set in the operation of union of sets, and the universal set is the identity set in the operation of intersection of sets.
20. For any two sets A and B, prove that A ∪ B = A ∩ B ⇔ A = B.
Sol: Given: A=B
Hence, A∪B =A∪A=A and A∩B=A∩A=A. It implies that A=B ⇒ A ∪ B=A ∩ B.
Suppose, A ∪ B=A ∩ B.
Let x ∈ A ∪ B; it implies that x ∈ A ∩ B
So, x ∈ A and x ∈ B
For every x, this belongs to both A and B Therefore, A=B
Try yourself:
20. Let A,B,C be any three sets such that A∪B=A∪C and A∩B=A∩C. Then, show that B=C.
1.5.3 Commutative Law
Commutative law for union: For any two sets AandB the commutative law for union holds good; it means that A ∪ B=B ∪ A
Commutative law for intersection: For any two sets AandB the commutative law for intersection holds good; it means that
A ∩ B=B ∩ A.
Commutative law for difference of sets: For any two sets AandB the commutative law for difference of sets may not hold; it means that A–B ≠ B–A
Commutative law for symmetric difference of sets: For any two sets AandB the commutative law for symmetric difference holds good; it means that A ∆ B =B ∆ A.
21. If A = {1, 2, 3, 4, 6} and B = {3, 4, 5, 7, 9}, show that intersection is commutative.
Sol: Consider A ∩ B = {3, 4} and B ∩ A = {3, 4}.
Therefore, A ∩ B=B ∩ A
Try yourself:
21. If A = {1, 2, 3, 4, 6} and B = {3, 4, 5, 7, 9}, show that A ∆ B =B ∆ A.
1.5.4 Associative Law
The associative law holds good in the operations of union of sets, intersection of sets, and symmetric difference of two sets.
1. For any three sets A,B,C
i. A ∪ (B ∪ C)=(A ∪ B) ∪ C
ii. A ∩ (B ∩ C)=(A ∩ B) ∩ C
iii. A ∆ (B ∆ C)=(A ∆ B) ∆ C
2. Associative law does not hold in the operation of difference on sets.
A–(B–C) ≠ (A–B)–C
22. Prove that A–(B–C) = (A–B) ∪ (A ∩ C).
Sol: A–(B–C) = A–(B ∩ C')
= A ∩ (B ∩ C')'
= A∩ (B' ∪ C) = (A ∩ B') ∪ (A ∩ C)
= (A–B) ∪ (A ∩ C)
Try yourself:
22. If A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, and C = {4, 5, 6, 7}, then prove that A–(B–C) ≠ (A–B)–C.
1.5.5 Distributive Law
If A,BandC are any three sets, the union and intersection are distributive over intersection and union, respectively.
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
Try yourself:
23. If A = {1, 2, 4, 5}, B = {2, 3, 5, 6}, and C = {4, 5, 6, 7}, then prove that A ∩ (B ∆ C) = (A ∩ B) ∆(A∩ C).
1.5.6 De Morgan’s Law
Let A,BandC be any three sets.
i. (A ∪ B)ʹ = Aʹ ∩ Bʹ
ii. (A ∩ B)ʹ = Aʹ ∪ Bʹ
23. Prove that ()()()() ABABABBA ∪−∩=−∪−
Sol: Consider: (A ∪ B) – A ∩ B ()() ()() ()() () ()()() ()()()() ()() ()() ABAB ABAB ABAB ABAABB AABAABBB BAAB ABBA ∪−∩ ′ =∪∩∩ ′′=∪∩∪ ′′=∪∩∪∪∩ ′′′′ =∩∪∩∪∩∪∩ ′′=∩∪∩ =−∪− ∴()()()() ABABABBA ∪−∩=−∪−
Try yourself:
24. Prove that (A ∩ B) – C = (A – C) ∩ (B – C).
TEST YOURSELF
1. For any two sets A and B, A ∩ (A ∪ B) = (1) A (2) B (3) ϕ (4) none of these
2. For any two sets A and B, (A–B) ∪ (B–A) = (1) (A−B) ∪ A (2) (B−A) ∪ B (3) (A ∪ B) – (A ∩ B) (4) (A ∪ B) ∩ (A ∩ B)
3. Which of the following statements is false?
(1) A−B = A ∩ B'
(2) A−B = A – (A ∩ B)
(3) A−B = A–B'
(4) A−B =( A ∪ B) – B
4. The set ()() ABCABCC ′ ′ ∪∩′∪∩∩′ ∩ is equal to
(1) B ∩ Cʹ
(2) A ∩ C
(3) B ∪ Cʹ
(4) A ∩ Cʹ
Answer Key
(1) 1 (2) 3 (3) 3 (4) 1
1.6 RESULTS ON CARDINAL NUMBER OF SETS
Cardinal number of a set is equal to the number of elements in that set. It will not give the nature of the elements of set. In this topic, we will discuss the cardinal number of elements of union or intersection or difference of two or more sets.
1.6.1 Cardinal Number of Union or Intersection or Difference of Two Sets
Let A,B, and C be any three finite sets and U be the finite universal set in this context.
i. n(A ∪ B) = n(A) + n(B) – n(A ∩ B)
ii. n(A ∩ B) = n(A) + n(B) – n(A ∪ B)
iii. If AandB are disjoint sets, then n(A ∪ B) = n(A) + n(B)
iv. n(A' ∪ B') = n(U) – n(A ∩ B)
v. n(A' ∩ B') n(U) – n(A ∪ B)
vi. n(A–B) = n(A) – n(A ∩ B)
vii. n(A–B) + n(A ∩ B) = n(A)
viii. n(A ∆ B) = n(A) + n(B) – 2n(A ∩ B)
Consider the Venn diagram
A s p r q U
Observing the Venn diagram,
(i) n(A) = p+r,n(B) = q+r
(ii) n(A ∪ B) = p+q+r
(iii) n(A') = s–p–r,n(B’) = s–q–r
(iv) n(A' ∪ B' ) = s–r,n(A' ∩ B') = s–p–q–r
(v) n(A–B) = p,n(B–A) = q
(vi) n(A ∩ B) = r, n(A ∆ B) = p+q
24. For any two finite sets A and B show that ()()()() 2 nABnAnBnAB ∆=+−∩
Sol: Given: AandB are two finite sets.
We know that the symmetric difference of two sets is
()() ABABBA ∆=−∪−
Consider the cardinal number of elements on both sides
()()() () ()() nABnABBA nABnBA ∆=−∪− =−+−
Because A–B,B–A are two disjoint sets. Hence, ()()()()() ()()() 2 nABnAnABnBnAB nAnBnAB ∆=−∩+−∩ =+−∩
Therefore, n(A ∆B) = n(B) + n(B) – 2n(A ∩ B)
Try yourself:
25. If A and B are disjoint sets with n ( A ) = 3 and n(B) = 6, then n{(A–B) ∪ (B–A)} = ?
Ans: 9
1.6.2 Cardinal Number of Union and Intersection of Three Sets
Let A,B, and C be any three sets.
(i) Number of elements in at least one of the three sets A,B,C is
n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C )
– n ( A ∩ B ) – n ( B ∩ C ) – n ( C ∩ A ) + n(A ∩ B ∩ C)
(ii) Number of elements in exactly two of the three sets A,B,andC is
n ( A ∩ B ) + n ( B ∩ C ) + n ( C ∩ A ) –3n(A ∩ B ∩ C)
(iii) Number of elements in exactly one of the three sets A,B,andC is
n(A) + n(B) + n(C) – 2n(A∩B) –2n(B∩C)
– 2n(C ∩ A) + 3n(A ∩ B ∩ C)
(iv) If n(A) = a,n(B) = b,n(C)=c,n(A∩B) = d, n(B∩C) = e,n(C∩A) = f, n(A∩B∩C) = g then the Venn diagram is as shown below.
R5. (A ∩ B' ∩ C'), i.e., A only
R6. (A' ∩ B ∩ C'), i.e., B only
R7 (A' ∩ B' ∩ C), i.e., C only
R8 (A' ∩ B' ∩ C') = () ABC ∪∪ ; neither A nor B nor C
25. In a school, there are 50 students taking examinations in Mathematics, Physics, and Chemistry. Each of the students has passed in at least one of the subjects, 37 passed Mathematics, 24 passed Physics, and 43 passed Chemistry. At most, 19 passed Mathematics and Physics; at most, 29 passed Mathematics and Chemistry; and, at most, 20 passed Physics and Chemistry. Then, find the largest possible number of students that could have passed all the three examinations.
Sol: Given: () 50, nMPC∪∪= and ()()()37,24,43nMnPnC===
Given: ()()() 19,20, 29 nMPnPCnCM ∩≤∩≤∩≤
We know that () ()()()() ()()() nMPC nMnPnCnMP nPCnCPnMPC ∪∪ =++−∩ −∩−∩+∩∩
Substitute the values.
(v) Consider the Venn diagram.
R5 R2 R1 R6 R7 R8 R3 R4
■ The numbers in the above diagram represent below as sets.
R1. A ∩ B ∩ C
R2 (A ∩ B ∩ C')
R3 (A' ∩ B ∩ C)
R4. (A ∩ B' ∩ C)
50 ≥ 37 + 24 + 43 – 19 – 20 –29 + n(M ∩ P ∩ C)
Hence, n(M ∩ P ∩ C) ≤ 14
Therefore, the largest possible number of students that could have passed all three examinations is 14.
Try yourself:
26. In a class of 55 students, the number of students studying different subjects is 23 in Mathematics, 24 in Physics, 19 in Chemistry, 12 in Mathematics and Physics, 9 in Mathematics and Chemistry, 7 in Physics and Chemistry, and 4 in all the three subjects. Find the number of students who have taken exactly one subject.
Ans: 22
TEST YOURSELF
1. If A and B are two sets such that n(A) = 70, n(B) = 60, and n(A∪B) =110, then n(A∩B) = (1) 240 (2) 50 (3) 40 (4) 20
2. If n ( U ) = 60, n ( A ) = 35, n ( B ) = 24, and n(A ∪ B)' = 10, then n(A ∩ B) is (1) 9 (2) 8 (3) 6 (4) 7
3. Let U be the universal set for sets A and B If n(A) = 200, n(B) = 300, and n(A ∩ B) = 100, then n(A'∩B') is equal to 300, provided that n(U) is equal to
(1) 600 (2) 700 (3) 800 (4) 900
4. In a committee, 50 people speak French, 20 speak Spanish, and 10 speak both Spanish and French. The number of persons speaking at least one of these two languages is
(1) 60 (2) 40 (3) 38 (4) 22
5. Each student, in a class of 40, studies at least one of the subjects English, Mathematics, and Economics. 16 study English, 22 study Economics, 26 study Mathematics, 5 study English and Economics, 14 study Mathematics and Economics, and 2 study all the three subjects. The number of students who study English and Mathematics but not Economics is (1) 7 (2) 5 (3) 10 (4) 4
6. In a college of 300 students, every student reads 5 newspapers and every newspaper is read by 50 students. The number of newspapers is (1) at least 30 (2) at most 30 (3) exactly 30 (4) none of these
7. Two newspapers A and B are published in a city. It is known that 25% of the city’s population reads A and 20% reads B, while
8% reads both A and B . Further, 30% of those who read A but not B look into advertisements and 40% of those who read B but not A also look into advertisements, while 50% of those who read both A and B look into advertisements. Then, the percentage of the population who look into advertisements is (1) 12.8 (2) 13.9 (3) 13 (4) 15
8. In a city, 20% of the population travels by car, 50% travels by bus, and 10% travels by both car and bus. Then, the percentage of persons travelling by car or bus is (1) 80% (2) 40% (3) 60% (4) 70%
9. An investigator interviewed 100 students to determine the performance of three drinks: milk, coffee, and tea. The investigator reported that 10 students take all three drinks, 20 students take milk and coffee, 25 students take milk and tea, 20 students take coffee and tea, 12 students take milk only, 5 students take coffee only, and 8 students take tea only. Then, the number of students who do not take any of the three drinks is
(1) 10 (2) 20 (3) 25 (4) 30
10. Each set X r contains 5 elements, each set Y r contains 2 elements, and 20 11 n rr rr XSY == ==
If each element of S belongs to exactly 10 of the X’s and to exactly 4 of the Y’s, then n is
(1) 10 (2) 20 (3) 100 (4) 50
Answer Key
(1) 4 (2) 1 (3) 2 (4) 1 (5) 2 (6) 3 (7) 2 (8) 3
(9) 4 (10) 2
CHAPTER REVIEW
Definition and representation of sets
1. A set is a collection of well-defined objects which are distinct from each other. Sets are generally denoted by capital letters.
2. If a is an element of the set A then we write it as ∈ aA , if a is not an element of the set A then we write it as ∉ aA
3. A set can be represented in two ways
4. Roster method, in this method a set is described by listing elements, separated by commas and enclosing them by curly brackets
5. Set Builder form: In this method, we write down a property or rule or proposition, which gives us all the element of the set.
6. Cardinal number of a finite set: The number of elements in a finite set A is called the cardinality of the set A and is denoted by ()nA . It is also called the cardinal number of the set.
Types of sets
7. Null set or Empty set: A set having no element in it is called an empty set or null set or void set. It is denoted by φ or {}
8. A set consisting of at least one element is called a non-empty set or a non-void set.
9. Singleton set: A set which has only finite number of elements is called a finite set.
10. Infinite set: A set which has an infinite number of elements is called an infinite set.
11. Universal set: A set consisting of all possible elements which occur in the discussion is called a Universal set and is denoted by U . All sets are contained in the universal set.
12. Comparability of sets:
Let A and B be two sets, if every element of A is an element of B , then A is called a subset of B. If A is subset B then ⊆ AB
If A is subset of B and ≠ AB then A is proper subset of B and we write ⊂ AB
• Every set is subset of it self
• Empty set is a subset of every set
• The total number of subsets of a finite set containing n elements is 2n
Let A be any set. The set of all subsets of A is called power set of A and is denoted by P(A)
• Power set of empty set contains only one element.
• Power set of a given set is always nonempty set.
Operations on sets
13. Union of two sets: It contains all the elements of both sets.
∪=∈∈ :or ABxxAxB
{}
14. Intersection of two sets: It contains all the elements which belong to both the sets.
∩=∈∈ :and ABxxAxB
{}
15. If the intersection of two sets is empty set then those two sets are called disjoint sets.
∩=∈∈ :and ABxxAxB
{}
16. Difference of two sets: The difference of two sets is defined as
{}−=∈∉ :and ABxxAxB
17. Compliment of a set is defined as {} =∈∉ :and C AxxUxA
18. The symmetric difference of two sets is defined as A ∆ B = {x: x ∈ A and x ∉ B or x ∉ A and x ∈ B}
19. Venn diagrams:
Laws of Algebra of sets
20. Commutative law:
• ∪=∪ ABBA
• ∩=∩ ABBA
21. Associative law
• ()() ∪∪=∪∪ ABCABC
• ()() ∩∩=∩∩ ABCABC
22. Distributive laws
(a)
23. De-Morgan's law
(a) (A ∪ B)' = A' ∩ B'
B
(b) (A ∩ B)' = A' ∪ B'
23. Results on Cardinal number of sets
If A, B,and C are finite sets U be the finite universel set then,
• n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B )
• n ( A ∪ B ) = n ( A ) + n ( B ) ⇔ A,B are disjoint sets
• n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n( A ∩ B ) – n( B ∩ C ) – n( A ∩ C ) + n ( A ∩ B ∩ C )
Exercises
JEE MAIN LEVEL
Level – I
Definition and Representation of a Set
Single Option Correct MCQs
1. Which of the following is a set?
(1) Collection of flowers in a village
(2) Collection of names of people in the world
(3) Collection of natural numbers below 20
(4) All of the above
2. The collection of intelligent students in a class is
(1) a null set
(2) a singleton set
(3) a finite set
(4) not a set
3. Which of the following is not a set?
(1) set of prime numbers
(2) set of beautiful places in India
(3) set of even numbers
(4) set of squares of all odd numbers
4. Represent the given set in roster form:
A = {x : x3 – 2x2 – x + 2 = 0, x ∈ R}
(1) A = {–1, 1, –2}
(2) A = {–1, 1, 2}
(3) A = {–1, 2}
(4) A = {1, 2}
5. Represent the given set in roster form:
A = {x : x2 – 8x + 15 = 0, x ∈ R}
(1) A = {3, 5} (2) A = {3, –5}
(3) A = {–3, 5} (4) A = {–3, –5}
6. Which of the following set builder forms is incorrect for the set E = {}?
(1) E = { x : x is a natural number and x < 1}
(2) E = {x : x is a prime number which is divisible by 3 and x < 100}
(3) E = {x : x is an odd number which is divisible by 2 and x < 50}
(4) E = {x : x is a whole number and x < 6}
7. Write the given set in roster form: {x ∈ N : x is a two-digit number such that the sum of its digits is either 6 or 9}
(1) {15, 24, 33, 51, 20, 18, 87, 54, 53, 72, 81, 99}
(2) {65, 24, 33, 42, 56, 60, 09, 18, 87, 36, 45, 64, 63, 72, 81, 90}
(3) {15, 24, 33, 42, 51, 60, 18, 27, 36, 45, 54, 63, 72, 81, 90}
(4) {26,25, 24, 34, 42, 51, 60, 18, 27, 36, 45, 54, 63, 72, 81, 90}
8. Represent the set 23456 ,,,, 72663124215
in the set - builder form.
(1) 3 :,16 1
∈≤≤ + n nNn n
(2) 3 :,26 1
∈≤≤ n nNn n
(3) 3 :,17 1 ∈≤≤ n nNn n
(4) 2 :,26 1 ∈≤≤ n nNn n
9. Write the given set in roster form: (x : x ∈ Z and –2 ≤ x ≤2}
(1) {–2, –1, 0, 1, 2} (2) {–1, 0, 1}
(3) {0, 1, 2} (4) None of these
10. Write the set 123456789 ,,,,,,,, 2345678910 in the set builder form.
(1) : , , 9 1
=∈< + n xxnNn n
(2) : , , 9 1
=∈≤ + n xxnNn n
(3) : , , 9 1
=∈≤ + n xxnZn n
(4) : , , 9 1
=∈< + n xxnZn n
Types of Sets
Single Option Correct MCQs
11. Among the following, an infinite set is
(1) the set of natural numbers ≤ 100
(2) the set of even numbers between 50 and 100
(3) the set of points on a circle
(4) the set of prime numbers between 10 and 50
12. Find the set of all natural numbers x such that 3x + 7 < 28.
(1) {1, 2, 3, 4, 5, 6}
(2) Infinite set
(3) {0, 1, 2, 3, 4, 5, 6}
(4) {1, 2, 3, 4, 5, 6, 7}
13. Write the interval (–7, 0) in the set builder form.
(1) {x : x ∈ R and –7 < x ≤ 0}
(2) {x : x ∈ R and –7 ≤ x ≤ 0}
(3) {x : x ∈ R and –7 < x < 0}
(4) {x : x ∈ R and –7 ≤ x < 0}
14. Write the set {x : x ∈ R and –4 < x ≤ 6} in interval form
(1) (–4, 6] (2) (–4, 6)
(3) [–4, 6] (4) {–4, 6}
15. A = { x : x is a letter in the word REAP}
Which of the following is an equal set
(1) B = {x : x is a letter in the word 'reep'}
(2) C = {x : x is a letter in the word 'rope'}
(3) D = {x : x is a letter in the word 'paper'}
(4) E = {x : x is a letter in the word 'pure'}
16. If A = { x : x – 5 = 0} then which of the following sets is equal to
(1) B = {x : x2 = 25}
(2) C = {x : x ≤ 5}
(3) E = {x : x is an integral positive root of the equation x2 – 2x – 15 = 0}
(4) F = {x : | x | = 5}
Comparability of Sets
Single Option Correct MCQs
17. If A = {1, 2, 3, 4, 5}, then the number of proper subsets of A is (1) 120 (2) 30 (3) 31 (4) 32
18. The number of non empty subsets of the set {1, 2, 3, 4} is (1) 14 (2) 15 (3) 16 (4) 17
19. If A = {1, 2, 3} then n(P(A)) = ? (1) 3 (2) 8 (3) 7 (4) 4
20. If A = {a,b} then the power set of A is (1) {ab , ba} (2) {a2 , b2} (3) { φ , {a}, {b}} (4) {φ, {a}, {b}, {a, b}}
21. If A is any set such that n[P(A)] = 64, then n(A) = (1) 32 (2) 16 (3) 8 (4) 6
22. If n(A) = 5, n[P(A)] = (1) 5 (2) 0 (3) 25 (4) 32
Operations on Sets
Single Option Correct MCQs
23. If A = {n/n is a digit in the number 33591} and B = {n/n ∈ N, n < 10}, then B – A = (1) {2, 4, 6, 8} (2) {7, 2, 4, 8, 6} (3) {1, 3, 5, 7} (4) {(1, 2), (1, 3), (2, 3)}
24. If A is the set of the divisors of the number 15, B is the set of prime numbers smaller than 10, and C is the set of even numbers smaller than 9, then (A ∪ C) ∩ B is the set
(1) {1, 3, 5} (2) {1, 2, 3} (3) {2, 3, 5} (4) {2, 5}
25. Which of the following is correct, if P(A) is the power set of the set A?
(1) P(A) ∪ P(B) = P(A ∪ B)
(2) P(A) ∩ P(B) = P(A ∩ B)
(3) P(A) – P(B) = P(A – B)
(4) None of these
26. If A = {1, 3, 5, 7, 9, 11, 13, 15, 17}, B = {2, 4, ......., 18}, and N is the universal set, then A' ∪ ((A ∪ B) ∩ B') is (1) A (2) N (3) B (4) R
27. Let A = {x : x ∈ R, x > 4} and B = {x : x ∈ R, x < 5}. Then, A ∩ B = (1) (4, 5] (2) (4, 5) (3) [4, 5) (4) [4, 5]
28. If V = {a, e, i, o, u} and B = {a, i, k, u}, then V – B = (1) {a, e, i, o, u} (2) {a, e, o, u} (3) {e, o} (4) {a, e, o}
29. Let A , B and C be finite sets such that A ∩ B ∩ C = φ and each one of the sets A ∆ B, B ∆ C, and C ∆ A has 100 elements. The number of elements in A ∪ B ∪ C = (1) 250 (2) 200 (3) 150 (4) 300
Laws of Algebra of Sets
Single Option Correct MCQs
30. If A and B are two given sets, then A ∩ (A ∩ B)C is equal to (1) A (2) B (3) φ (4) A ∩ BC
31. If A ∩ B = B, then (1) A ⊆ B (2) B ⊆ A
(3) A = φ (4) B = φ
32. For any two sets A and B, A ∩ (A ∪ B)c is equal to (1) A (2) B
(3) φ (4) A ∩ B
33. The set (A ∪ B)c ∪ (B ∩ C) is equal to (1) Ac ∪ B ∪ C (2) Ac ∩ C
(3) Ac ∪ C ' (4) Ac ∩ B
Results on Cardinal Number of Sets
Single Option Correct MCQs
34. In a city, three daily newspapers A, B, and C are published. 42% of the people in that city read A, 51% read B, 68% read C, 30% read A and B, 28% read B and C, 36% read A and C, 8% do not read any of the three newspapers. The percentage of persons who read all the three papers is
(1) 25% (2) 18%
(3) 20% (4) 30%
35. In a group of 1000 people, each may speak either Hindi or English or both. There are 750 people who can speak Hindi and 400 who can speak English. Then, the number of persons who can speak only Hindi is
(1) 300 (2) 400
(3) 600 (4) 450
36. In a town of 10,000 families it was found that 40% families, buy newspaper A, 20% families buy newspaper B, 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C, and 4% buy A and C. If 2% families buy all the three newspapers, then the number of families that buy only A is
(1) 3100 (2) 3300
(3) 2900 (4) 1400
37. A survey shows that 63% of the Americans like cheese whereas 76% like apples. If x% of the Americans like both cheese and apples, then
(1) x = 39 (2) x = 63
(3) 39 ≤ x ≤ 63 (4) 30 ≤ x ≤ 80
38. Out of 800 boys in a school, 224 played cricket, 240 played hockey, and 336 played basketball. Of the total, 64 played both basketball and hockey 80 played cricket and basketball, 40 played cricket and hockey and 24 played all the three games. The number of boys who did not play any game is (1) 160 (2) 240 (3) 216 (4) 128
39. A survey of 500 television viewers produced the following information: 285 watch football, 195 watch hockey, 115 watch basketball, 45 watch football and basketball, 70 watch football and hockey, and 50 watch hockey and basketball, 50 do not watch any of the three games. The number of viewers, who watch exactly one of the three games, is
(1) 325 (2) 310 (3) 315 (4) 372
40. In a class of 60 students, 23 play hockey, 15 play basket ball and 20 play cricket. 7 play hockey and basket Ball, 5 play cricket and basketball, 4 play hockey and cricket, and 15 students do not play any of these games. Then
(1) 4 play hockey, basket ball and cricket (2) 20 play hockey but not cricket (3) 1 plays hockey and cricket but not basketball (4) all the above are correct
41. A school awarded 38 medals in football, 15 in basketball and 20 in cricket. Suppose these medals went to a total of 58 students and only three students got medals in all three sports. If only 5 students got medals in football and basketball, then the number of medals received in exactly two of three sports is
(1) 7 (2) 9
(3) 11 (4) 13
42. Find the value of n(A ∪ B) using the given Venn diagram.
(3) 40 (4) 48
43. A set A has 3 elements and another set B has 6 elements. Then,
(1) 3 ≤ n(A ∪ B) ≤ 6
(2) 3 ≤ n(A ∪ B) ≤ 9
(3) 6 ≤ n(A ∪ B) ≤ 9
(4) 0 ≤ n(A ∪ B) ≤ 9
44. If the difference between the number of subsets of the sets A and B is 120, then choose the incorrect option.
(1) Maximum value of n(A ∩ B) = 3
(2) Minimum value of n(A ∩ B) = 0
(3) Maximum value of n(A ∪ B) = 21
(4) Minimum value of n(A ∪ B) = 7
45. In a statistical investigation of 1,003 families of Kolkata, it was found that 63 families have neither a radio nor a TV. 794 families have a radio, and 187 have a TV. Find the number of families in that group having both A radio and a TV.
(1) 43 (2) 37 (3) 41 (4) 47
46. In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics, 70 study Chemistry, 40 study Mathematics and Physics, 30 study Physics and Chemistry, 50 study Chemistry and Mathematics, and 20 student none of these subjects. Find the number of students who study all the three subjects.
(1) 30 (2) 20 (3) 40 (4) 35
47. Let μ be the universal set containing 700 elements. If A and B are subsets of μ such that n(A) = 200, n(B) = 300 and n(A ∩ B) = 100. Then n(A' ∩ B') =
(1) 400 (2) 600 (3) 300 (4) 500
48. In a cla ss, 18 students offered physics, 23 offered chemistry and 24 offered Mathematics. Of these, 13 are in both chemistry and Mathematics, 12 in Physics and Chemistry, 11 in Physics and Mathematics, and 6 in all the three subjects.
Statement I : The number of students offered Mathematics but not Chemistry = 11
Statement II : The number of students who offered exactly one of the three subjects = 11
In light of the above statements, choose the correct answer from the options given below.
(1) Statement I is true and Statement II is false
(2) Statement I is false and Statement II is true
(3) Statement I and statement II both are true
(4) Statement I and statement II both are false
49. In the Venn diagram given below, find the value of a, if n(X ∪ Y) = 45.
X Y 20 8 a
(1) 11 (2) 25 (3) 17 (4) 20
Numerical Value Questions
50. Let X = {n ∈ N : 1 ≤ n ≤ 50}. If A = {n ∈ X : n is a multiple of 2} and B = {n ∈ X : n is a multiple of 7}, then the number of elements in the smallest subset of X containing both A and B is
51. In a battle, 70% of the combatants lost one eye, lost 80% an ear, 75% lost an arm, 85% lost a leg, and x % lost all the four limbs. The minimum values of x is
52. In a class of 35 students, 17 have taken Mathematics, and 10 have taken Mathematics but not Economics. If each student has taken either Mathematics or Economics or both, then the number of students who have taken Economics but not Mathematics is
Level – II
Types of Sets
Single Option Correct MCQs
1. The number of elements in the set.
{(a, b)/2a2 + 3b2 = 35, a, b ∈ z} when z is the set of all integers is (1) 2 (2) 4 (3) 8 (4) 12
2. Which of the following is infinite set?
(1) {x : x ∈ N and x is prime}
(2) {x : x ∈ N and x is an even number less than 20}
(3) {x : x2 = 4 and x is an odd integer}
(4) {x : x is an even prime greater than 2}
Comparability of Sets
Single Option Correct MCQs
3. If P(x) represents the power set of ‘x’, then n[P[P[P( φ ]]] = (1) 0 (2) 1 (3) 2 (4) 4
Operations on Sets
Single Option Correct MCQs
4. Consider the following equations
a) A – B = A – (A ∩ B)
b) A = (A ∩ B) ∪ (A – B)
c) A – (B ∪ C) = (A – B) ∪ (A – C)
Which of these is/are correct?
(1) a and c
(2) b only
(3) b and c
(4) a and b
5. Suppose that A 1, A 2, A 3, ...... A 50 are fifty sets, each containing 6 elements and B 1, B2, B3, ......B n are n sets, each containing 3 elements. If 50 11 == == n i j ij UASUB and each element of S belongs to exactly 10 of the Ai ’s and exactly 9 of the B j’s then n =
(1) 45 (2) 54 (3) 90 (4) 18
6. Let S be the set of points inside the square, T be the set of points inside the triangle, and C be the set of points inside the circle. If the triangle and circle intersect each other and are contained in a square, then
(1) S ∩ T ∩ C = φ
(2) S ∪ T ∪ C = C
(3) S ∪ T ∪ C = S
(4) S ∪ T = S ∩ C
7. Let F 1 be the set of all parallelograms, F 2 be the set of all rectangles, F 3 be the set of all rhombuses, F4 be the set of all squares, and F 5 be the set of trapeziums in a plane. Then, F1 may be equal to
(1) F2 ∩ F
Laws of Algebra of Sets
Single Option Correct MCQs
8. Which is simplified representation of (A' ∩ B' ∩ C) ∪ (B ∩ C) ∪ (A ∩ C) where A, B, C are subsets of set X?
(1) A
(2) B
(3) C
(4) X ∩ (A ∪ B ∪ C)
Results on Cardinal Number of Sets
Single Option Correct MCQs
9. In a certain town 25% families own a phone and 15% own a car, 65% families own neither a phone nor a car, 2000 families own both a car and a phone. Consider the following statements in this regard.
A) 10% families own both a car and a phone
B) 35% families own either a car or a phone
C) 40,000 families live in the town.
Which of the following statements are correct
(1) A and B (2) A and C
(3) B and C (4) A, B and C
Numerical Value Questions
10. A market research group conducted a survey of 1000 consumers and reported that 720 consumers liked product A and 450 consumers liked product B. The least number they must have liked both products is .
THEORY-BASED QUESTIONS
Very Short Answer Questions
1. Write the set builder form of {2, 5, 8, 11,.....}.
2. Write in roster form {x / x = 3(2)n–1 , n = 1, 2, 3,....}
3. Write the symbols used between an element and a set.
4. What is the definition of equivalent sets?
5. What is the number of proper subsets of a singleton set?
6. What is the improper subset of a non-empty set?
7. What is the maximum possible number of elements of union of two sets A, B?
8. For any two sets A and B, what is the set builder form of A∆B?
9. If n(A) = p and n(B) = q, then find the range of n(A ∪ B).
10. If n(A) = p and n(B) = q, then find the range of n(A ∩ B).
11. If there are three sets A,B, and C, then write the expression for the number of elements in exactly one of the given three sets.
12. If there are three sets A,B, and C, then write the expression for the number of elements in exactly two of the given three sets.
13. Let A and B be two subsets of a universal set U and n(A) = p, n(B) = q,n(A ∩ B) = r, n ( U ) = s . Then what is the expression of n(A' ∩ B')?
Statement Type Questions
Each question has two statements: statement I (S-I) and statement II (S-II). Mark the correct answer as
(1) if both statement I and statement II are correct,
(2) if both statement I and statement II are incorrect,
(3) if statement I is correct, but statement II is incorrect,
(4) if statement I is incorrect, but statement II is correct.
14. S-I : Collection of names of students of a class who are born in the month of May is a well defined collection.
S-II : Collection of names of good students in a class is not well defined.
15. S-I : All equal sets are equivalent sets.
S-II : All equivalent sets are equal sets.
16. S-I : The symmetric difference of A and B is (A ∪ B) – (A ∩ B).
S-II : The symmetric difference of A and B is (A–B) ∪ (B–A).
17. Let A and B be any two sets and P(X) be the power set of X
S-I : P(A ∪ B) = P(A) ∪ P(B)
S-II : P(A ∩ B) = P(A) ∩ P(B)
18. S-I : If n(A) = 3 and n(B) = 5, then the minimum possible number of elements in the set A ∪ B is 5.
S-II : If n(A) = p and n(B) = q, then the range of n ( A ∪ B ) is [ p,q ] where p <q.
19. S-I : If A and B are two disjoint sets, then A–B=A.
S-II : If A and B are two disjoint sets, then B–A=B.
20. S-I : If A is a sub set of B, then B is the superset of A.
S-II : If A is a proper subset of B, then all the elements of A are the elements of B and there exists at least one element in B which is not in A.
21. S-I : If A ∩ B = φ, then the sets A and B are called disjoint sets.
S-II : If A ∩ B ≠ φ, then the sets A and B are said to be intersecting or overlapping sets.
22. S-I : If A ∩ B is an empty set, then n(A–B) = n(A).
S-II : n(A–B) = n(A) – n(A ∩ B).
Assertion and Reason Questions
In each of the following questions, a statement of Assertion (A) is given, followed by a corresponding statement of Reason (R). Mark the correct answer as
(1) if both (A) and (R) are true and (R) is the correct explanation of (A),
(2) if both (A) and (R) are true but (R) is not the correct explanation of (A),
(3) if (A) is true but (R) is false,
(4) if both (A) and (R) are false.
23. (A) : Set builder form of 1234 ,,, 2345 is :,5, 1 n xxnnN n =<∈ +
(R) : Each set can be represented in set builder form in a unique way.
24. (A) : The number of elements in the power set of an empty set is 1.
(R) : The number of subsets of a set having n elements is 2n
25. (A) : If n ( A ) = 3 and n ( B ) = 5, then the maximum possible number of elements in the set A ∩ B is 3.
(R) : If n ( A ) = p and n ( B ) = q, then the range of n(A ∩ B) is [0, p] where p < q.
26. (A) : If A = {x / x ∈ N, x is a prime number} and B = {x / x ∈ N}, then (A ∩ B) = A
BRAIN TEASERS
1 Let
Then, sum of the values of n(S) is equal to (1) 4 (2) 0 (3) 6 (4) 2
FLASHBACK (P revious JEE Q uestions )
JEE Main
1. Let A and B be two finite sets with m and n elements, respectively. The total number of subsets of the set A is 56 more than the total number of subsets of B. Then, the distance of the point P(m, n) from the point Q(–2, –3) is (27th Jan 24 Shift 2) (1) 10 (2) 6 (3) 4 (4) 8
2. A group of 40 students appeared in an examination of 3 subjects – Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students
(R) : If A ⊆ B then A ∩ B=A.
27. (A) : If a set A has 4 elements then the number of elements in the power set of A is 14.
(R) : If n ( A ) = n , the number of proper subsets of A is 2n
28. (A) : n(A ∆ B) = n(A) + n(B)–2n(A ∩ B)
(R) : A ∆ B=A ∪ B–A ∩ B
29. (A) : If A and B are two non-empty sets and U is the universal set in this context, then n(A' ∩ B') = n(U)–n(A ∪ B).
(R) : A' ∩ B' = (A ∪ B)'
30. (A) : If A and B are any two disjoint sets, then n(A ∪ B) = n(A) + n(B).
(R) : n(A∩B) = 0 when A and B are disjoint sets.
2. A and B are two non-empty sets and A is a proper subset of B . If n ( A ) = 4, then minimum possible value of n ( A Δ B ) is (where Δ denotes symmetric difference of sets A and B).
passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students that passed in all the three subjects is ________.
(30th Jan 24 Shift 1)
M P C x
3. An organisation awarded 48 medals in event ‘A’, 25 in event ‘B’, and 18 in event ‘C’. If these medals went to total 60 men and only five men got medals in all the three events, then how many received medals in exactly two of three events? (11th Apr 23 Shift 1)
(1) 21 (2) 10 (3) 15 (4) 9
4. Let A = {x ∈ R : | x + 1 | < 2 } and B = {x ∈ R : |x – 1| ≥ 2}.
CHAPTER TEST – JEE MAIN
Section - A
1. Consider two sets A and B such that A ∪ (B ∩ {1, 2, 3}) = {2, 3, 4, 5}. Then, which of the following is correct?
(1) 1 ∈ A ∪ B
(2) 1 ∈ A' ∩ B'
(3) Minimum value of n(A) is 4.
(4) minimum value of n(B) is 2.
2. In a certain town, 25% families own a cell phone, 15% families own a scooter, and 65% families own neither a cell phone nor a scooter. If 1500 families own both a cell phone and a scooter, then the total number of families in the town is (1) 10000 (2) 20000 (3) 30000 (4) 40000
3. Suppose, A 1 , A 2 , A 3 , ……. A 30 are thirty sets, each with 5 elements, and B1, B2, B3, B n are n sets, each with three elements, such that 30 11 n ij ij ABS == ==∪∪ . If each element of S belongs to exactly 10 of the Ai's and exactly 9 of the Bj's, then the value of n is
(1) 15 (2) 135 (3) 45 (4) 90
4. If U = {2, 3, 5, 7, 9} is the universal set and A = {3, 7}, B = {2, 5, 7, 9}, then find A'∪ B'.
Then which one of the following statements is NOT true? (25th Jun 22 Shift 2)
(1) A–B=(–1,1)
(2) B–A = R – (–3, 1)
(3) A ∩ B = (–3, –1]
(4) A ∪ B = R –[1, 3)
5. The number of elements in the set {n∈N :10 ≤ n ≤ 100 and 3n – 3 is a multiple of 7} is ______. (15th Apr 23 Shift 1)
(1) A' ∪ B' = {2, 3, 5, 9}
(2) A' ∪ B' ={ 2, 3, 9}
(3) A' ∪ B' = {2, 3, 5, 7, 9}
(4) A' ∪ B' = {2, 5, 9}
5. Let S = {1, 2, 3.......100}. The number of non-empty subsets A of S such that the product of elements in A is even is
(1) 250–1 (2) 250 (250–1)
(3) 2100–1 (4) 250 + 1
6. If A = {5n/n ∈ N, n≤100}, B = {3n/n ∈ N, n≥ 100}, and C = {2n/n ∈ N}, then the number of elements in (A ∩ B) ∩ C' is
(1) 7 (2) 13 (3) 21 (4) 22
7. In a survey, it was found out that 100 people don’t use any of laptop, mobile or wrist watches. 80 persons use all the three gadgets. There are 150 who use laptop and mobile, 200 who use mobile and wristwatch and 200 who use laptop and wristwatch. The number of people who use only laptop, only mobile and only wristwatch is equal. If this survey was conducted on 1000 persons, how many people use only wristwatch?
(1) 200 (2) 170 (3) 150 (4) 900
8. If A = {x:x = 4n + 1, n ≤ 5, n ≤ N} and B = {3n : n ≤ 8, n ≤ N}, then find A – (A–B)?
(1) {5} (2) {7}
(3) {9} (4) {11}
9. A set contains 2n + 1 elements. The number of subsets of this set containing more than n elements equals to (1) 2n–1 (2) 2n (3) 2n+1 (4) 22n
10. Two sets A and B are as follows:
A = {(a, b) ∈ R × R : |a − 5| < 1, |b − 5| < 1}
B = {(a, b) ∈ R × R : 4 (a − 6)2 + 9(b − 5)2 ≤ 36} Then (1) A ⊂ B
(2) A ∩ B = ϕ (an empty set)
(3) neither A ⊂ B nor B ⊂ A (4) B ⊂ A
11 If U is the universal set and A ∪ B ∪ C = U, then [(A–B) ∪ (B – C) ∪ (C – A)]' equals (1) A ∪ B ∪ C (2) A ∩ B ∩ C (3) A ∪( B ∩ C) (4) A ∩( B ∪ C)
12. If A = {x ∈ C : x2 = 1} and B = {x ∈ C: x4 = 1}, then find A ∆ B.
(1) {–1, 1} (2) {–1, 1, i,–i} (3) {–i,i} (4) {–1, i}
13. If A and B are two sets, then (A ∪ B)c ∪ (Ac ∩ B )= (1) Ac (2) Bc (3) ϕ (4) ∪
14. A survey shows that 53% of the people in a city read a magazine A whereas 69% read a magazine B. If x% of the people read both the magazines then a possible value of x can be
(1) 76 (2) 55 (3) 45 (4) 65
15.I f n(A)=50, n(B)=20 and n(A ∩ B) =10, then n(AΔB) = k2+1. Then, k =
(1) ±5 (2) ±7 (3) ±6 (4) ±8
16. In a class of 140 students, numbered 1 to 140, all even numbered students opted for Mathematics course, those whose number is divisible by 3 opted for Physics course and those whose number is divisible by 5 opted for Chemistry course. Then the number of students who did not opt for any of the three courses is
(1) 102 (2) 42 (3) 1 (4) 38
17. Of the members of three athletic teams in a school, 21 are in the cricket team, 26 are in the hockey team, and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is (1) 43 (2) 76 (3) 49 (4) 53
18. If n ( U ) = 60, n ( A ) = 21, n ( B ) = 43 then greatest value of n ( A ∪ B ) and least value of n(A ∪ B) are
(1) 60, 43 (2) 50, 36 (3) 70, 44 (4) 60, 38
19. In a class of 100 students, 55 students have passed in Mathematics and 67 students have passed in Physics. The number of students who have passed only in Physics is (1) 512 (2) 43 (3) 54 (4) 45
20. If A= {prime numbers less than 30}, B={natural numbers less than 10}, then find (A–B) ∩ (B–A)
(1) {11, 13, 17, 19, 23, 29}
(2) {1, 2, 3, 4, 5, 6, 7, 8, 9}
(3) φ
(4) A ∩ B
Section - B
21. There is a group of 265 persons who like either singing or dancing or painting. In this group, 200 like singing, 110 like dancing, and 55 like painting. If 60 persons like both singing and dancing, 30 like both singing and painting, and 10 like all the three activities, then the number of persons who like only dancing and painting is ___.
22. Let X be the set consisting of the first 2018 terms of the arithmetic progression 1,6,11, …….. and Y be the set consisting of the first 2018 terms of the arithmetic progression 9, 16, 23, ………… Then, the number of elements in the set X ∪ Y is ___.
23. 90 students take Mathematics and 72 students take Science in a class of 120 students. If 10 take neither Mathematics nor Science, then the number of students who take both the subjects is ____.
24. In a survey of 400 students in a school, 100 were listed as taking apple juice, 150 as taking orange juice, and 75 were listed as taking both apple as well as orange juice. Then, how many students were taking neither apple juice nor orange juice?
ANSWER KEY
JEE Main Level
25. In a school, on the Republic Day, three dramas A,B, and C are performed on the dais. In a group of people who attended the function and who like at least one of the three dramas, 16 people like A, 20 people like B, 15 people like C, 4 people like both A and B, 3 people like both A and C, 3 people like both B and C and 2 people like all the three dramas. Then, how many people like at most two dramas?
Theory-based Questions
(4) Two or more sets having same cardinal number are equivalent (5) 1 (6) It self (7)
Brain Teasers
Flashback
Chapter Test – JEE Main
