Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add.

- Manish Nagar Branch, Beltarodi Road Nagpur - 440037

Mathematics SETS

SETS

Key Concepts

1. A set is a well-defined collection of objects.

2. Sets can be represented in two ways—Roster or tabular form and Set builder form.
3. Roster form: All the elements of a set are listed and separated by commas and are
enclosed withinbraces { }. Elements are not repeated generally.
4. Set builder form: In set builder form, a set is denoted by stating the properties that
its memberssatisfy.
5. A set does not change if one or more elements of the set are repeated.
6. An empty set is the set having no elements in it. It is denoted by ϕ or { }.
7. A set having a single element is called a singleton set.
8. On the basis of number of elements, sets are of two types—Finite and Infinite sets.
9. A finite set is a set in which there are a definite number of elements. Now, ϕ or { } or null
set is a finiteset as it has 0 number of elements, which is a definite number.
10. A set that is not finite is called an infinite set.
11. All infinite sets cannot be described in the roster form.
12. Two sets are equal if they have the exactly same elements.
13. Two sets are said to be equivalent if they have the same number of elements.
14. Set A is a subset of set B if every element of A is in B, i.e. there is no element in A which
is not in Band is denoted by A ⊂ B.
15. A is a proper subset of B if and only if every element in A is also in B and there exists at
least oneelement in B that is not in A.
16. If A is a proper subset of B, then B is a superset of A and is denoted by B ⊃ A.
17. Let A be a set. Then, the collection of all subsets of A is called the power set of A and is
denoted by P(A).

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS
18. Common set notations
N: The set of all natural numbers
Z: The set of all integers
Q: The set of all rational numbers
R: The set of real numbers
Z+: The set of positive integers
Q+: The set of positive rational numbers
R+: The set of positive real numbers
N  R , Q  R , Q  Z, R  Z and N  R+

19. Two sets are equal if A  B and B  A, then A = B.

20. Null set  is subset of every set including the null set itself.
21. The set of all the subsets of A is known as the power set of A.
22. Open interval: The interval which contains all the elements between a and b excluding
a and b. Inset notations:
(a, b) ={ x : a < x < b}

Closed interval: The interval which contains all the elements between a and b and also the
endpoints a and b is called the closed interval.
[a, b] = {x : a ≤ x ≤ b}

23. Semi-open intervals

[a, b) = {x : a ≤ x < b} includes all the elements from a to b including a and excluding b

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS
(a, b] = {x : a ≤ x < b} includes all the elements from a to b excluding a and including b.

24. Universal set refers to a particular context.

It is the basic set that is relevant to that context. The universal set is usually denoted by U.
25. Union of sets A and B, denoted by A  B is defined as the set of all the elements which
are either inA or B or both.
26. Intersection of sets A and B, which are denoted by A  B, is defined as the set of all the
elements,which are common to both A and B.
27. The difference of the sets A and B is the set of elements, which belong to A but not to
B and it iswritten as A − B and read as ‘A minus B’.
In set notations A − B = {x : xA, xB} and B − A = {x: xB, xA}.
28. If the intersection of two non-empty sets is empty, i.e. A ∩ B = , then A and B are
disjoint sets.
29. Let U be the universal set and A be a subset of U. Then the complement of
A, writtenas A’ or Ac, is the set of all elements of U that are not in set A.
30. The number of elements present in a set is known as the cardinal number of the set or
cardinality ofthe set. It is denoted by n(A).
31. If A is a subset of U, then A’ is also a subset of U.
32. Counting theorems are together known as Inclusion–Exclusion Principle. It helps in
determining thecardinality of union and intersection of sets.
33. Sets can be represented graphically using venn diagrams. Venn diagrams consist of
rectangles andclosed curves, usually circles. The universal set is generally represented by
a rectangle and its subsets by circles.

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS
Key Formulae

1. Union of sets A  B ={x:x A or x B }

2. Intersection of sets A  B ={x:x A and x B }

3. Complement of a set A’ = {x: xU and xA}, A’ = U – A

4. Difference of sets A − B = {x: xA, xB} and B − A = {x: xB, xA}

5. Symmetric difference of two sets: Let A and B be two sets. Then the symmetric difference
of sets A - B   B - A and it is denoted by AB
AB = (A - B)  (B - A) = {x : x  A, x  B, x  A  B}

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS

6. Properties of the operation of union

a. Commutative law:
b. A  B = B  A
c. Associative law:
(A  B)  C = A  (B  C)
d. Law of identity:
e. A   = A
f. Idempotent law:
g. A  A = A
h. Law of U  A = U

6. Properties of operation of intersection

i) Commutative law:
AB=BA
ii) Associative law:
iii) (A  B)  C = A  (B  C)
iv) Law of  and U:
  A =  and U  A = U
v) Idempotent law:
AA=A
vi) Distributive law:
A  (B  C) = (A  B)  (A  C)

7. Properties of complement of sets

a. Complement laws:
i. A  A’ = U
ii. A  A’ = 
b. De-Morgan’s law:
i. (A  B)’ = A’  B’
ii. (A  B)’ = A’  B’
iii. Law of double Complementation:

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS
(A’)’ = A
c. Laws of empty set and universal set:
’ = U and U’ = 

8. Operations on sets
i. A - B = A  B'
ii. B - A = A' B
iii. A -B = A  A  B = 
iv. (A – B)  B = A  B
v. (A - B)  B = 
vi. A  B  B'  A'
vii. (A – B)  (B -A) = (A  B) - (A  B)

9. Some more important results: Let A, B and C be three sets. Then, we have
i. A – (B  C) = (A - B)  (A - C)
ii. A – (B  C) = (A - B)  (A - C)
iii. A  (B - C) = (A  B) - (A  C)
iv. A  (B  C) = (A  B)  (A  C)

10. Counting Theorems

a. If A and B are finite sets, then the number of elements in the union of two sets
is given byn(A  B ) = n(A) + n(B) − n(A  B)
b. If A and B are finite sets and A  B = 
then n(A  B) = n(A) + n(B)
c. n(A  B) = n(A − B) + n(B − A) + n(A  B)
d. n(A  B  C) = n(A) + n(B) + n(C) − n(B  C) − n(A  B) − n(A  C) + n(A  B  C)
e. Number of elements in exactly two of the sets =
n(A  B) + n(B  C) + n(C  A) -3n(A  B  C)
f. Number of elements in exactly one of the sets =
=n(A) + n(B) + n(C) – 2n(A  B) – 2n(B  C) -2n (C  A) + 3n(A  B  C)
g. n(A’B’) = n((A B)’)= n(U) – n(A B)
h. n(A’ B’) = n((A B)’)= n(U) – n(A B)

11. Number of elements in the power set of a set with n elements =2n.
Number of proper subsets in the power set = 2n − 1.

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 MATHS SETS

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

Important Questions
Multiple Choice questions-
Question 1. If f(x) = log [(1 + x)/(1 – x), then f(2x )/(1 + x²) is equal to
(a) 2f(x)
(b) {f(x)}²
(c) {f(x)}³
(d) 3f(x)
Question 2. The smallest set a such that A ∪ {1, 2} = {1, 2, 3, 5, 9} is
(a) {3, 5, 9}
(b) {2, 3, 5}
(c) {1, 2, 5, 9}
(d) None of these
Question 3. Let R= {(x, y): x, y belong to N, 2x + y = 41}. The range is of the relation R is
(a) {(2n – 1) : n belongs to N, 1 ≤ n ≤ 20}
(b) {(2n + 2) : n belongs to N, 1 < n < 20}
(c) {2n : n belongs to N, 1< n< 20}
(d) {(2n + 1) : n belongs to N , 1 ≤ n ≤ 20}
Question 4. Empty set is a?
(a) Finite Set
(b) Invalid Set
(c) None of the above
(d) Infinite Set
Question 5. 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. The values of M and N are
respectively.
(a) 6, 3
(b) 8, 5
(c) None of these
(d) 4, 1
Question 6. If the number of elements in a set S are 5. Then the number of elements of the
power set P(S) are?

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

(a) 5
(b) 6
(c) 16
(d) 32
Question 7. Every set is a ___________ of itself
(a) None of the above
(b) Improper subset
(c) Compliment
(d) Proper subset
Question 8. If x ≠ 1, and f(x) = x + 1 / x – 1 is a real function, then f(f(f(2))) is
(a) 2
(b) 1
(c) 4
(d) 3
Question 9. In 3rd Quadrant?
(a) X < 0, Y < 0 (b) X > 0, Y < 0
(c) X < 0, Y > 0
(d) X < 0, Y > 0
Question 10. IF A ∪ B = A ∪ C and A ∩ B = A ∩ C, THEN
(a) none of these
(b) B = C only when A I C
(c) B = C only when A ? B
(d) B = C
Very Short Questions:
1. The collection of all the months of a year beginning with letter M
2. The collection of difficult topics in Mathematics.
Let A = {1,3,5,7,9}. Insert the appropriate symbol Î or Ï in blank spaces: – (Question- 3,4)
3. 2 ------- A
4. 5 -------- A
5. Write the set A = {x: x is an integer, –1 ≤ x < 4} in roster form

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

6. List all the elements of the set,

A = {x: x ∈ Z, − < x < }
7. Write the set B = {3,9,27,81} in set-builder form
8. A = {x : x ∈ N and 3 <x <4}
9. B = {x : x ∈ N and x2 = x}
Short Questions:
1. In a group of 800 people, 500 can speak Hindi and 320 can speak English. Find
(i) How many can speak both Hindi and English?
(ii) How many can speak Hindi only?
2. A survey shows that 84% of the Indians like grapes, whereas 45% like pineapple. What percentage
of Indians like both grapes and pineapple?
3. In a survey of 450 people, it was found that 110 play crickets, 160 play tennis and 70 play
both cricket as well as tennis. How many play neither cricket nor tennis?
4. In a group of students, 225 students know French, 100 know Spanish and 45 know both.
Each student knows either French or Spanish. How many students are there in the group?
5. If A = - 3, 5), B = (0, 6 then find (i) A – B, (ii) A ∪ B
6. 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. Find how many
students were taking neither apple juice nor orange juice.
Long Questions:
1. In a survey it is found that 21 people like product A, 26 people like product B and 29 like
product C. If 14 people like product A and B, 15 people like product B and C, 12 people like
product C and A, and 8 people like all the three products. Find
(i) How many people are surveyed in all?
(ii) How many like product C only?
2. A college awarded 38 medals in football, 15 in basket ball and 20 in cricket. If these medals
went to a total of 50 men and only five men got medals in all the three sports, how many
received medals in exactly two of the three sports?
3. There are 200 individuals with a skin disorder, 120 had been exposed to the chemical C 1, 50
to chemical C2, and 30 to both the chemicals C1 and C2. Find the number of individuals
exposed to
(1) chemical C1 but not chemical C2
(2) chemical C2 but not chemical C1

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

(3) chemical C1 or chemical C2

4. In a survey it was found that 21 peoples liked product A, 26 liked product B and 29 liked
product C. If 14 people liked products A and B, 12 people like C and A, 15 people like B and
C and 8 liked all the three products. Find now many liked product C only
5. A college awarded 38 medals in football, 15 in basketball and 20 in cricket. If these medals
went to a total of 58 men and only three men got medal in all the three sports, how many
received medals in exactly two of the three sports?
Assertion Reason Questions:
1. In each of the following questions, a statement of Assertion is given followed by a
corresponding statement of Reason just below it. Of the statements, mark the correct
answer as.
Assertion (A) : ‘The collection of all natural numbers less than 100’ is a set.
Reason (R) : A set is a well-defined collection of the distinct objects.
(i) Both assertion and reason are true and reason is the correct explanation of assertion.
(ii) Both assertion and reason are true but reason is not the correct explanation of
assertion.
(iii) Assertion is true but reason is false.
(iv) Assertion is false but reason is true.
2. In each of the following questions, a statement of Assertion is given followed by a
corresponding statement of Reason just below it. Of the statements, mark the correct
answer as.
Assertion (A) : The set D = {x : x is a prime number which is a divisor of 60} in roster form is
{1, 2, 3, 4, 5}.
Reason (R) : The set E = the set of all letters in the word ‘TRIGONOMETRY’, in the roster
form is {T, R, I, G, O, N, M, E, Y}.
(i) Both assertion and reason are true and reason is the correct explanation of assertion.
(ii) Both assertion and reason are true but reason is not the correct explanation of
assertion.
(iii) Assertion is true but reason is false.
(iv) Assertion is false but reason is true.

Answer Key:
MCQ
1. (a) 2f(x)

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

2. (a) {3, 5, 9}
3. (a) {(2n – 1) : n belongs to N, 1 ≤ n ≤ 20}
4. (a) Finite Set
5. (a) 6, 3
6. (d) 32
7. (b) Improper subset
8. (d) 3
9. (a) X < 0, Y < 0
10.(d) B = C
Very Short Answer:
1. Set
2. 2. Not a set
3. 3. ∉
4. 4. ∈
5. 5. A = {–1, 0, 1, 2, 3}
6. 6. A = {0,1,2,3,4,5}
7. 7. B = {x: x = 3n, n ∈ N and 1 ≤ n ≤ 4}
8. 8. Empty set
9. 9. Non-empty set
Short Answer:
Ans: 1. (i) 20 people can speak both Hindi and English
(ii) 480 people can speak Hindi only
Ans: 2. 29% of the Indians like both grapes and pineapple.
Ans: 3. ∪ – set of people surveyed
A – set of people who play cricket
B – set of people who play tennis
Number of people who play neither cricket nor tennis
= n (A ∪ B)′′= n(U) – n(A ∪ B)
= 450 – 200
= 250

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

Ans: 4. (i) -3, 0; (ii) -3,6

Ans: 5. Let A denote the set of students taking apple juice and B denote the set of students taking
orange juice
n(U) = 400, n(A) = 100, n (B) = 150 n(AB) =75
n ( (A′ ∩ B′) ) = n(A ∪ B)′
= n(∪) – n(A′ ∪ B′)
= n(∪) – [n(A) + n(B) – n(A ∩ B)]
=400 – 100 – 150 + 75 = 225
Long Answer:
Ans: 1. Let A, B, C denote respectively the set of people who like product
A, B, C.

a, b, c, d, e, f, g – Number of elements in bounded region

(i) Total number of Surveyed people = a + b + c + d + e + f + g = 43
(ii) Number of people who like product C only = g = 10
Ans: 2. people got medals in exactly two of the three sports.

f=5
a + b + f + e = 38
b + c + d + f = 15
e + d + f + g = 20
a + b + c + d + e + f + g = 50
Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

we have to find b + d + e
Ans: 3. A denote the set of individuals exposed to the chemical C 1 and B denote the set of
individuals exposed to the chemical C2
n(U) = 200, n(A) = 120, n(B) = 50, n(AB) = 30
(i) n(A-B) = n(A) – n(AB)
=120-30=90
(ii)n(B-A) = n(B) – n(AB)
=50-30 = 20
(iii)n(A∪B) =n(A)+ n(B)-n(AB)
=120+50-30
=140
Ans: 4. a + b + c + d = 21
b + c + e + f = 26
c + d + f + g = 29

b + c = 14,c + f =15,c + d = 12
c=8
d = 4, c = 8,f = 7,b = 6,g = 10,e = 5,a = 3
like product c only = g = 10
Ans: 5. Let A, B and C denotes the set of men who received medals in football, basketball and
cricket respectively.
n (A) = 38, n (B) = 15, n (C) = 20
n (A B C) = 58 and n (A ∩ B ∩ C) = 3

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037
 SETS

n (A B C) = n (A) + n (B) + n(C) – n (A ∩ B) – n (B ∩ C) – n (C ∩ A) + n (A ∩ B ∩ C)

58 = 38 + 15 + 20 – (a + d ) – (d + c) – (b + d) + 3
18 = a + d + c + b + d
18 = a + b + c + 3d
18 = a + b + c + 3 3
9=a+b+c
Assertion Reason Answer:
1. (i) Both assertion and reason are true and reason is the correct explanation of assertion.
2. (iv) Assertion is false but reason is true.

Chinmay S. Bhagat (Career Counsellor) - 9403413415 , 07123589249 , Add. - Manish Nagar Branch, Beltarodi Road Nagpur - 440037