Class 11
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NCERT Solutions
Sets
49 Solutions
Exercise:
All Exercises
EXERCISE 1.1
EXERCISE 1.2
EXERCISE 1.3
EXERCISE 1.4
EXERCISE 1.5
Miscellaneous Exercise on Chapter 1
Q1
EXERCISE 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.
Q2
EXERCISE 1.1
Let
A
=
{
1
,
2
,
3
,
4
,
5
,
6
}
\mathrm{A}=\{1,2,3,4,5,6\}
A
=
{
1
,
2
,
3
,
4
,
5
,
6
}
. Insert the appropriate symbol
∈
\in
∈
or
∉
\notin
∈
/
in the blank spaces:
(i)
5
…
A
5 \dots \mathrm{A}
5
…
A
(ii)
8
…
A
8 \dots \mathrm{A}
8
…
A
(iii)
0
…
A
0 \dots \mathrm{A}
0
…
A
(iv)
4
…
A
4 \dots \mathrm{A}
4
…
A
(v)
2
…
A
2 \dots \mathrm{A}
2
…
A
(vi)
10
…
A
10 \dots \mathrm{A}
10
…
A
Q3
EXERCISE 1.1
Write the following sets in roster form:
(i)
A
=
{
x
:
x
\mathrm{A}=\{x: x
A
=
{
x
:
x
is an integer and
−
3
≤
x
<
7
}
-3 \leq x<7\}
−
3
≤
x
<
7
}
(ii)
B
=
{
x
:
x
\mathrm{B}=\{x: x
B
=
{
x
:
x
is a natural number less than 6}
(
i
i
i
)
(iii)
(
iii
)
\mathrm{C}={x: x
i
s
a
t
w
o
−
d
i
g
i
t
n
a
t
u
r
a
l
n
u
m
b
e
r
s
u
c
h
t
h
a
t
t
h
e
s
u
m
o
f
i
t
s
d
i
g
i
t
s
i
s
8
}
is a two-digit natural number such that the sum of its digits is 8\}
i
s
a
tw
o
−
d
i
g
i
t
na
t
u
r
a
l
n
u
mb
ers
u
c
h
t
ha
tt
h
es
u
m
o
f
i
t
s
d
i
g
i
t
s
i
s
8
}
(iv)
D
=
{
x
:
x
\mathrm{D}=\{x: x
D
=
{
x
:
x
is a prime number which is divisor of 60}
(
v
)
(v)
(
v
)
\mathrm{E}=
T
h
e
s
e
t
o
f
a
l
l
l
e
t
t
e
r
s
i
n
t
h
e
w
o
r
d
T
R
I
G
O
N
O
M
E
T
R
Y
(
v
i
)
The set of all letters in the word TRIGONOMETRY (vi)
T
h
ese
t
o
f
a
lll
e
tt
ers
in
t
h
e
w
or
d
TR
I
GONOMETR
Y
(
v
i
)
\mathrm{F}=$ The set of all letters in the word BETTER
Q4
EXERCISE 1.1
Write the following sets in the set-builder form :
(i)
(
3
,
6
,
9
,
12
)
(3,6,9,12)
(
3
,
6
,
9
,
12
)
(ii)
{
2
,
4
,
8
,
16
,
32
}
\{2,4,8,16,32\}
{
2
,
4
,
8
,
16
,
32
}
(iii)
{
5
,
25
,
125
,
625
}
\{5,25,125,625\}
{
5
,
25
,
125
,
625
}
(iv)
{
2
,
4
,
6
,
…
}
\{2,4,6, \dots\}
{
2
,
4
,
6
,
…
}
(v)
{
1
,
4
,
9
,
…
,
100
}
\{1,4,9, \dots, 100\}
{
1
,
4
,
9
,
…
,
100
}
Q5
EXERCISE 1.1
List all the elements of the following sets :
(i)
A
=
{
x
:
x
\mathrm{A}=\{x: x
A
=
{
x
:
x
is an odd natural number}
(
i
i
)
(ii)
(
ii
)
\mathrm{B}={x: x
i
s
a
n
i
n
t
e
g
e
r
,
is an integer,
i
s
anin
t
e
g
er
,
-\frac{1}{2}<x<\frac{9}{2}}
(
i
i
i
)
(iii)
(
iii
)
\mathrm{C}={x: x
i
s
a
n
i
n
t
e
g
e
r
,
is an integer,
i
s
anin
t
e
g
er
,
x^{2} \leq 4}
(
i
v
)
(iv)
(
i
v
)
\mathrm{D}={x: x
i
s
a
l
e
t
t
e
r
i
n
t
h
e
w
o
r
d
"
L
O
Y
A
L
"
}
is a letter in the word "LOYAL"\}
i
s
a
l
e
tt
er
in
t
h
e
w
or
d
"
L
O
Y
A
L
"
}
(v)
E
=
{
x
:
x
\mathrm{E}=\{x: x
E
=
{
x
:
x
is a month of a year not having 31 days}
(
v
i
)
(vi)
(
v
i
)
\mathrm{F}={x: x
i
s
a
c
o
n
s
o
n
a
n
t
i
n
t
h
e
E
n
g
l
i
s
h
a
l
p
h
a
b
e
t
w
h
i
c
h
p
r
e
c
e
d
e
s
is a consonant in the English alphabet which precedes
i
s
a
co
n
so
nan
t
in
t
h
e
E
n
g
l
i
s
ha
lp
hab
e
tw
hi
c
h
p
rece
d
es
k}$
Q6
EXERCISE 1.1
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
}
\{1,2,3,6\}
{
1
,
2
,
3
,
6
}
(ii)
{
2
,
3
}
\{2,3\}
{
2
,
3
}
(iii)
{M,A,T,H,E,I,C,S}
(iv)
{
1
,
3
,
5
,
7
,
9
}
\{1,3,5,7,9\}
{
1
,
3
,
5
,
7
,
9
}
(a)
{
x
:
x
\{x: x
{
x
:
x
is a prime number and a divisor of 6}
(b)
{
x
:
x
\{x: x
{
x
:
x
is an odd natural number less than 10}
(c)
{
x
:
x
\{x: x
{
x
:
x
is natural number and divisor of 6}
(d)
{
x
:
x
\{x: x
{
x
:
x
is a letter of the word MATHEMATICS}
Q1
EXERCISE 1.2
Which of the following are examples of the null set
(i)
Set of odd natural numbers divisible by 2
(ii)
Set of even prime numbers
(iii)
{
x
:
x
\{x: x
{
x
:
x
is a natural numbers,
x
<
5
x<5
x
<
5
and
x
>
7
}
x>7\}
x
>
7
}
(iv)
{
y
:
y
\{y: y
{
y
:
y
is a point common to any two parallel lines}
Q2
EXERCISE 1.2
Which of the following sets are finite or infinite
(i)
The set of months of a year
(ii)
{
1
,
2
,
3
,
…
}
\{1,2,3, \dots\}
{
1
,
2
,
3
,
…
}
(iii)
{
1
,
2
,
3
,
…
99
,
100
}
\{1,2,3, \dots 99, 100\}
{
1
,
2
,
3
,
…
99
,
100
}
(iv)
The set of positive integers greater than 100
(v)
The set of prime numbers less than 99
Q3
EXERCISE 1.2
State whether each of the following set is finite or infinite:
(i)
The set of lines which are parallel to the
x
x
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
)
(0,0)
(
0
,
0
)
Q4
EXERCISE 1.2
In the following, state whether
A
=
B
\mathrm{A}=\mathrm{B}
A
=
B
or not:
(i)
A
=
{
a
,
b
,
c
,
d
}
B
=
{
d
,
c
,
b
,
a
}
\mathrm{A}=\{a, b, c, d\} \quad \mathrm{B}=\{d, c, b, a\}
A
=
{
a
,
b
,
c
,
d
}
B
=
{
d
,
c
,
b
,
a
}
(ii)
A
=
{
4
,
8
,
12
,
16
}
B
=
{
8
,
4
,
16
,
18
}
\mathrm{A}=\{4, 8, 12, 16\} \quad \mathrm{B}=\{8, 4, 16, 18\}
A
=
{
4
,
8
,
12
,
16
}
B
=
{
8
,
4
,
16
,
18
}
(iii)
A
=
{
2
,
4
,
6
,
8
,
10
}
B
=
{
x
:
x
\mathrm{A}=\{2, 4, 6, 8, 10\} \quad \mathrm{B}=\{x: x
A
=
{
2
,
4
,
6
,
8
,
10
}
B
=
{
x
:
x
is positive even integer and
x
≤
10
}
x \leq 10\}
x
≤
10
}
(iv)
A
=
{
x
:
x
\mathrm{A}=\{x: x
A
=
{
x
:
x
is a multiple of 10},
B
=
{
10
,
15
,
20
,
25
,
30
,
…
}
\quad \mathrm{B}=\{10, 15, 20, 25, 30, \dots\}
B
=
{
10
,
15
,
20
,
25
,
30
,
…
}
Q5
EXERCISE 1.2
Are the following pair of sets equal ? Give reasons.
(i)
A
=
{
2
,
3
}
,
B
=
{
x
:
x
\mathrm{A}=\{2,3\}, \quad \mathrm{B}=\{x: x
A
=
{
2
,
3
}
,
B
=
{
x
:
x
is solution of
x
2
+
5
x
+
6
=
0
}
x^{2}+5 x+6=0\}
x
2
+
5
x
+
6
=
0
}
(ii)
A
=
{
x
:
x
\mathrm{A}=\{x: x
A
=
{
x
:
x
is a letter in the word FOLLOW}
B
=
{
y
:
y
\mathrm{B}=\{y: y
B
=
{
y
:
y
is a letter in the word WOLF}
Q6
EXERCISE 1.2
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\} \nE=\{-1,1\}, & F=\{0, a\}, & G=\{1,-1\}, & H=\{0,1\} \end{array}$$
Q1
EXERCISE 1.3
Make correct statements by filling in the symbols
⊂
\subset
⊂
or
⊄
\not\subset
⊂
in the blank spaces :
(i)
{
2
,
3
,
4
}
…
{
1
,
2
,
3
,
4
,
5
}
\{2,3,4\} \dots \{1,2,3,4,5\}
{
2
,
3
,
4
}
…
{
1
,
2
,
3
,
4
,
5
}
(ii)
{
a
,
b
,
c
}
…
{
b
,
c
,
d
}
\{a, b, c\} \dots \{b, c, d\}
{
a
,
b
,
c
}
…
{
b
,
c
,
d
}
(iii)
{
x
:
x
\{x: x
{
x
:
x
is a student of Class XI of your school}
…
{
x
:
x
\dots \{x: x
…
{
x
:
x
student of your school}
(iv)
{
x
:
x
\{x: x
{
x
:
x
is a circle in the plane}
…
{
x
:
x
\dots \{x: x
…
{
x
:
x
is a circle in the same plane with radius 1 unit}
(v)
{
x
:
x
\{x: x
{
x
:
x
is a triangle in a plane}
…
{
x
:
x
\dots \{x: x
…
{
x
:
x
is a rectangle in the plane}
(vi)
{
x
:
x
\{x: x
{
x
:
x
is an equilateral triangle in a plane}
…
{
x
:
x
\dots \{x: x
…
{
x
:
x
is a triangle in the same plane}
(vii)
{
x
:
x
\{x: x
{
x
:
x
is an even natural number}
…
{
x
:
x
\dots \{x: x
…
{
x
:
x
is an integer}
Q2
EXERCISE 1.3
Examine whether the following statements are true or false:
(i)
{
a
,
b
}
⊄
{
b
,
c
,
a
}
\{a, b\} \not\subset \{b, c, a\}
{
a
,
b
}
⊂
{
b
,
c
,
a
}
(ii)
{
a
,
e
}
⊂
{
x
:
x
\{a, e\} \subset \{x: x
{
a
,
e
}
⊂
{
x
:
x
is a vowel in the English alphabet}
(iii)
{
1
,
2
,
3
}
⊂
{
1
,
3
,
5
}
\{1,2,3\} \subset \{1,3,5\}
{
1
,
2
,
3
}
⊂
{
1
,
3
,
5
}
(iv)
{
a
}
⊂
{
a
,
b
,
c
}
\{a\} \subset \{a, b, c\}
{
a
}
⊂
{
a
,
b
,
c
}
(v)
{
a
}
∈
{
a
,
b
,
c
}
\{a\} \in \{a, b, c\}
{
a
}
∈
{
a
,
b
,
c
}
(vi)
{
x
:
x
\{x: x
{
x
:
x
is an even natural number less than 6}
⊂
{
x
:
x
\subset \{x: x
⊂
{
x
:
x
is a natural number which divides 36}
Q3
EXERCISE 1.3
Let
A
=
{
1
,
2
,
{
3
,
4
}
,
5
}
\mathrm{A}=\{1,2,\{3,4\}, 5\}
A
=
{
1
,
2
,
{
3
,
4
}
,
5
}
. Which of the following statements are incorrect and why?
(i)
{
3
,
4
}
⊂
A
\{3,4\} \subset \mathrm{A}
{
3
,
4
}
⊂
A
(ii)
{
3
,
4
}
∈
A
\{3,4\} \in \mathrm{A}
{
3
,
4
}
∈
A
(iii)
{
{
3
,
4
}
}
⊂
A
\{\{3,4\}\} \subset \mathrm{A}
{{
3
,
4
}}
⊂
A
(iv)
1
∈
A
1 \in \mathrm{A}
1
∈
A
(v)
1
⊂
A
1 \subset \mathrm{A}
1
⊂
A
(vi)
{
1
,
2
,
5
}
⊂
A
\{1,2,5\} \subset \mathrm{A}
{
1
,
2
,
5
}
⊂
A
(vii)
{
1
,
2
,
5
}
∈
A
\{1,2,5\} \in \mathrm{A}
{
1
,
2
,
5
}
∈
A
(viii)
{
1
,
2
,
3
}
⊂
A
\{1,2,3\} \subset \mathrm{A}
{
1
,
2
,
3
}
⊂
A
(ix)
ϕ
∈
A
\phi \in \mathrm{A}
ϕ
∈
A
(x)
ϕ
⊂
A
\phi \subset \mathrm{A}
ϕ
⊂
A
(xi)
{
ϕ
}
⊂
A
\{\phi\} \subset \mathrm{A}
{
ϕ
}
⊂
A
Q4
EXERCISE 1.3
Write down all the subsets of the following sets
(i)
{
a
}
\{a\}
{
a
}
(ii)
{
a
,
b
}
\{a, b\}
{
a
,
b
}
(iii)
{
1
,
2
,
3
}
\{1,2,3\}
{
1
,
2
,
3
}
(iv)
ϕ
\phi
ϕ
Q5
EXERCISE 1.3
Write the following as intervals :
(i)
{
x
:
x
∈
R
,
−
4
<
x
≤
6
}
\{x: x \in \mathrm{R},-4<x \leq 6\}
{
x
:
x
∈
R
,
−
4
<
x
≤
6
}
(ii)
{
x
:
x
∈
R
,
−
12
<
x
<
−
10
}
\{x: x \in \mathrm{R},-12<x<-10\}
{
x
:
x
∈
R
,
−
12
<
x
<
−
10
}
(iii)
{
x
:
x
∈
R
,
0
≤
x
<
7
}
\{x: x \in \mathrm{R}, 0 \leq x<7\}
{
x
:
x
∈
R
,
0
≤
x
<
7
}
(iv)
{
x
:
x
∈
R
,
3
≤
x
≤
4
}
\{x: x \in \mathrm{R}, 3 \leq x \leq 4\}
{
x
:
x
∈
R
,
3
≤
x
≤
4
}
Q6
EXERCISE 1.3
Write the following intervals in set-builder form :
(i)
(
−
3
,
0
)
(-3,0)
(
−
3
,
0
)
(ii)
[
6
,
12
]
[6,12]
[
6
,
12
]
(iii)
(
6
,
12
]
(6,12]
(
6
,
12
]
(iv)
[
−
23
,
5
)
[-23,5)
[
−
23
,
5
)
Q7
EXERCISE 1.3
What universal set(s) would you propose for each of the following :
(i)
The set of right triangles.
(ii)
The set of isosceles triangles.
Q8
EXERCISE 1.3
Given the sets
A
=
{
1
,
3
,
5
}
,
B
=
{
2
,
4
,
6
}
\mathrm{A}=\{1,3,5\}, \mathrm{B}=\{2,4,6\}
A
=
{
1
,
3
,
5
}
,
B
=
{
2
,
4
,
6
}
and
C
=
{
0
,
2
,
4
,
6
,
8
}
\mathrm{C}=\{0,2,4,6,8\}
C
=
{
0
,
2
,
4
,
6
,
8
}
, which of the following may be considered as universal set ( s ) for all the three sets
A
,
B
\mathrm{A}, \mathrm{B}
A
,
B
and C
(i)
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
\{0,1,2,3,4,5,6\}
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
(ii)
ϕ
\phi
ϕ
(iii)
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
\{0,1,2,3,4,5,6,7,8,9,10\}
{
0
,
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
,
10
}
(iv)
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
\{1,2,3,4,5,6,7,8\}
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
}
Q1
EXERCISE 1.4
Find the union of each of the following pairs of sets :
(i)
X
=
{
1
,
3
,
5
}
Y
=
{
1
,
2
,
3
}
\mathrm{X}=\{1,3,5\} \quad \mathrm{Y}=\{1,2,3\}
X
=
{
1
,
3
,
5
}
Y
=
{
1
,
2
,
3
}
(ii)
A
=
{
a
,
e
,
i
,
o
,
u
}
B
=
{
a
,
b
,
c
}
\mathrm{A}=\{a, e, i, o, u\} \quad \mathrm{B}=\{a, b, c\}
A
=
{
a
,
e
,
i
,
o
,
u
}
B
=
{
a
,
b
,
c
}
(iii)
A
=
{
x
:
x
\mathrm{A}=\{x: x
A
=
{
x
:
x
is a natural number and multiple of 3}
B
=
{
x
:
x
\mathrm{B}=\{x: x
B
=
{
x
:
x
is a natural number less than 6}
(
i
v
)
(iv)
(
i
v
)
\mathrm{A}={x: x
i
s
a
n
a
t
u
r
a
l
n
u
m
b
e
r
a
n
d
is a natural number and
i
s
ana
t
u
r
a
l
n
u
mb
er
an
d
1<x \leq 6}
\mathrm{B}={x: x
i
s
a
n
a
t
u
r
a
l
n
u
m
b
e
r
a
n
d
is a natural number and
i
s
ana
t
u
r
a
l
n
u
mb
er
an
d
6<x<10}
(
v
)
(v)
(
v
)
\mathrm{A}={1,2,3}, \mathrm{B}=\phi$
Q2
EXERCISE 1.4
Let
A
=
{
a
,
b
}
,
B
=
{
a
,
b
,
c
}
\mathrm{A}=\{a, b\}, \mathrm{B}=\{a, b, c\}
A
=
{
a
,
b
}
,
B
=
{
a
,
b
,
c
}
. Is
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
? What is
A
∪
B
\mathrm{A} \cup \mathrm{B}
A
∪
B
?
Q3
EXERCISE 1.4
If A and B are two sets such that
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
, then what is
A
∪
B
\mathrm{A} \cup \mathrm{B}
A
∪
B
?
Q4
EXERCISE 1.4
If
A
=
{
1
,
2
,
3
,
4
}
,
B
=
{
3
,
4
,
5
,
6
}
,
C
=
{
5
,
6
,
7
,
8
}
\mathrm{A}=\{1,2,3,4\}, \mathrm{B}=\{3,4,5,6\}, \mathrm{C}=\{5,6,7,8\}
A
=
{
1
,
2
,
3
,
4
}
,
B
=
{
3
,
4
,
5
,
6
}
,
C
=
{
5
,
6
,
7
,
8
}
and
D
=
{
7
,
8
,
9
,
10
}
\mathrm{D}=\{7,8,9,10\}
D
=
{
7
,
8
,
9
,
10
}
; find
(i)
A
∪
B
\mathrm{A} \cup \mathrm{B}
A
∪
B
(ii)
A
∪
C
\mathrm{A} \cup \mathrm{C}
A
∪
C
(iii)
B
∪
C
\mathrm{B} \cup \mathrm{C}
B
∪
C
(iv)
B
∪
D
\mathrm{B} \cup \mathrm{D}
B
∪
D
(v)
A
∪
B
∪
C
\mathrm{A} \cup \mathrm{B} \cup \mathrm{C}
A
∪
B
∪
C
(vi)
A
∪
B
∪
D
\mathrm{A} \cup \mathrm{B} \cup \mathrm{D}
A
∪
B
∪
D
(vii)
B
∪
C
∪
D
\mathrm{B} \cup \mathrm{C} \cup \mathrm{D}
B
∪
C
∪
D
Q5
EXERCISE 1.4
Find the intersection of each pair of sets of question 1 above.
Q6
EXERCISE 1.4
If
A
=
{
3
,
5
,
7
,
9
,
11
}
,
B
=
{
7
,
9
,
11
,
13
}
,
C
=
{
11
,
13
,
15
}
\mathrm{A}=\{3,5,7,9,11\}, \mathrm{B}=\{7,9,11,13\}, \mathrm{C}=\{11,13,15\}
A
=
{
3
,
5
,
7
,
9
,
11
}
,
B
=
{
7
,
9
,
11
,
13
}
,
C
=
{
11
,
13
,
15
}
and
D
=
{
15
,
17
}
\mathrm{D}=\{15,17\}
D
=
{
15
,
17
}
; find
(i)
A
∩
B
\mathrm{A} \cap \mathrm{B}
A
∩
B
(ii)
B
∩
C
\mathrm{B} \cap \mathrm{C}
B
∩
C
(iii)
A
∩
C
∩
D
\mathrm{A} \cap \mathrm{C} \cap \mathrm{D}
A
∩
C
∩
D
(iv)
A
∩
C
\mathrm{A} \cap \mathrm{C}
A
∩
C
(v)
B
∩
D
\mathrm{B} \cap \mathrm{D}
B
∩
D
(vi)
A
∩
(
B
∪
C
)
\mathrm{A} \cap(\mathrm{B} \cup \mathrm{C})
A
∩
(
B
∪
C
)
(vii)
A
∩
D
\mathrm{A} \cap \mathrm{D}
A
∩
D
(viii)
A
∩
(
B
∪
D
)
\mathrm{A} \cap(\mathrm{B} \cup \mathrm{D})
A
∩
(
B
∪
D
)
(ix)
(
A
∩
B
)
∩
(
B
∪
C
)
(\mathrm{A} \cap \mathrm{B}) \cap(\mathrm{B} \cup \mathrm{C})
(
A
∩
B
)
∩
(
B
∪
C
)
(x)
(
A
∪
D
)
∩
(
B
∪
C
)
(\mathrm{A} \cup \mathrm{D}) \cap(\mathrm{B} \cup \mathrm{C})
(
A
∪
D
)
∩
(
B
∪
C
)
Q7
EXERCISE 1.4
If
A
=
{
x
:
x
\mathrm{A}=\{x: x
A
=
{
x
:
x
is a natural number },
B
=
{
x
:
x
\mathrm{B}=\{x: x
B
=
{
x
:
x
is an even natural number }
C
=
{
x
:
x
\mathrm{C}=\{x: x
C
=
{
x
:
x
is an odd natural number }
a
n
d
and
an
d
\mathrm{D}={x: x
i
s
a
p
r
i
m
e
n
u
m
b
e
r
}
is a prime number \}
i
s
a
p
r
im
e
n
u
mb
er
}
, find
(i)
A
∩
B
\mathrm{A} \cap \mathrm{B}
A
∩
B
(ii)
A
∩
C
\mathrm{A} \cap \mathrm{C}
A
∩
C
(iii)
A
∩
D
\mathrm{A} \cap \mathrm{D}
A
∩
D
(iv)
B
∩
C
\mathrm{B} \cap \mathrm{C}
B
∩
C
(v)
B
∩
D
\mathrm{B} \cap \mathrm{D}
B
∩
D
(vi)
C
∩
D
\mathrm{C} \cap \mathrm{D}
C
∩
D
Q8
EXERCISE 1.4
Which of the following pairs of sets are disjoint
(i)
{
1
,
2
,
3
,
4
}
\{1,2,3,4\}
{
1
,
2
,
3
,
4
}
and
{
x
:
x
\{x: x
{
x
:
x
is a natural number and
4
≤
x
≤
6
}
4 \leq x \leq 6\}
4
≤
x
≤
6
}
(ii)
{
a
,
e
,
i
,
o
,
u
}
\{a, e, i, o, u\}
{
a
,
e
,
i
,
o
,
u
}
and
{
c
,
d
,
e
,
f
}
\{c, d, e, f\}
{
c
,
d
,
e
,
f
}
(iii)
{
x
:
x
\{x: x
{
x
:
x
is an even integer}
a
n
d
and
an
d
{x: x$ is an odd integer}
Q9
EXERCISE 1.4
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
}
\mathrm{A}=\{3,6,9,12,15,18,21\}, \mathrm{B}=\{4,8,12,16,20\}, \mathrm{C}=\{2,4,6,8,10,12,14,16\}, \mathrm{D}=\{5,10,15,20\}
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
\mathrm{A}-\mathrm{B}
A
−
B
(ii)
A
−
C
\mathrm{A}-\mathrm{C}
A
−
C
(iii)
A
−
D
\mathrm{A}-\mathrm{D}
A
−
D
(iv)
B
−
A
\mathrm{B}-\mathrm{A}
B
−
A
(v)
C
−
A
\mathrm{C}-\mathrm{A}
C
−
A
(vi)
D
−
A
\mathrm{D}-\mathrm{A}
D
−
A
(vii)
B
−
C
\mathrm{B}-\mathrm{C}
B
−
C
(viii)
B
−
D
\mathrm{B}-\mathrm{D}
B
−
D
(ix)
C
−
B
\mathrm{C}-\mathrm{B}
C
−
B
(x)
D
−
B
\mathrm{D}-\mathrm{B}
D
−
B
(xi)
C
−
D
\mathrm{C}-\mathrm{D}
C
−
D
(xii)
D
−
C
\mathrm{D}-\mathrm{C}
D
−
C
Q10
EXERCISE 1.4
If
X
=
{
a
,
b
,
c
,
d
}
\mathrm{X}=\{a, b, c, d\}
X
=
{
a
,
b
,
c
,
d
}
and
Y
=
{
f
,
b
,
d
,
g
}
\mathrm{Y}=\{f, b, d, g\}
Y
=
{
f
,
b
,
d
,
g
}
, find
(i)
X
−
Y
\mathrm{X}-\mathrm{Y}
X
−
Y
(ii)
Y
−
X
\mathrm{Y}-\mathrm{X}
Y
−
X
(iii)
X
∩
Y
\mathrm{X} \cap \mathrm{Y}
X
∩
Y
Q11
EXERCISE 1.4
If
R
\mathbf{R}
R
is the set of real numbers and
Q
\mathbf{Q}
Q
is the set of rational numbers, then what is
R
−
Q
\mathbf{R}-\mathbf{Q}
R
−
Q
?
Q12
EXERCISE 1.4
State whether each of the following statement is true or false. Justify your answer.
(i)
{
2
,
3
,
4
,
5
}
\{2,3,4,5\}
{
2
,
3
,
4
,
5
}
and
{
3
,
6
}
\{3,6\}
{
3
,
6
}
are disjoint sets.
(ii)
{
a
,
e
,
i
,
o
,
u
}
\{a, e, i, o, u\}
{
a
,
e
,
i
,
o
,
u
}
and
{
a
,
b
,
c
,
d
}
\{a, b, c, d\}
{
a
,
b
,
c
,
d
}
are disjoint sets.
(iii)
{
2
,
6
,
10
,
14
}
\{2,6,10,14\}
{
2
,
6
,
10
,
14
}
and
{
3
,
7
,
11
,
15
}
\{3,7,11,15\}
{
3
,
7
,
11
,
15
}
are disjoint sets.
(iv)
{
2
,
6
,
10
}
\{2,6,10\}
{
2
,
6
,
10
}
and
{
3
,
7
,
11
}
\{3,7,11\}
{
3
,
7
,
11
}
are disjoint sets.
Q1
EXERCISE 1.5
Let
U
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
,
A
=
{
1
,
2
,
3
,
4
}
,
B
=
{
2
,
4
,
6
,
8
}
U=\{1,2,3,4,5,6,7,8,9\}, A=\{1,2,3,4\}, B=\{2,4,6,8\}
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
}
\mathrm{C}=\{3,4,5,6\}
C
=
{
3
,
4
,
5
,
6
}
. Find (i)
A
′
\mathrm{A}^{\prime}
A
′
(ii)
B
′
\mathrm{B}^{\prime}
B
′
(iii)
(
A
∪
C
)
′
(\mathrm{A} \cup \mathrm{C})^{\prime}
(
A
∪
C
)
′
(iv)
(
A
∪
B
)
′
(\mathrm{A} \cup \mathrm{B})^{\prime}
(
A
∪
B
)
′
(v)
(
A
′
)
′
(\mathrm{A}^{\prime})^{\prime}
(
A
′
)
′
(vi)
(
B
−
C
)
′
(\mathrm{B}-\mathrm{C})^{\prime}
(
B
−
C
)
′
Q2
EXERCISE 1.5
If
U
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
}
\mathrm{U}=\{a, b, c, d, e, f, g, h\}
U
=
{
a
,
b
,
c
,
d
,
e
,
f
,
g
,
h
}
, find the complements of the following sets :
(i)
A
=
{
a
,
b
,
c
}
\mathrm{A}=\{a, b, c\}
A
=
{
a
,
b
,
c
}
(ii)
B
=
{
d
,
e
,
f
,
g
}
\mathrm{B}=\{d, e, f, g\}
B
=
{
d
,
e
,
f
,
g
}
(iii)
C
=
{
a
,
c
,
e
,
g
}
\mathrm{C}=\{a, c, e, g\}
C
=
{
a
,
c
,
e
,
g
}
(iv)
D
=
{
f
,
g
,
h
,
a
}
\mathrm{D}=\{f, g, h, a\}
D
=
{
f
,
g
,
h
,
a
}
Q3
EXERCISE 1.5
Taking the set of natural numbers as the universal set, write down the complements of the following sets:
(i)
{
x
:
x
\{x: x
{
x
:
x
is an even natural number}
(ii)
{
x
:
x
\{x: x
{
x
:
x
is an odd natural number}
(iii)
{
x
:
x
\{x: x
{
x
:
x
is a positive multiple of 3}
(iv)
{
x
:
x
\{x: x
{
x
:
x
is a prime number}
(v)
{
x
:
x
\{x: x
{
x
:
x
is a natural number divisible by 3 and 5}
(vi)
{
x
:
x
\{x: x
{
x
:
x
is a perfect square}
(vii)
{
x
:
x
\{x: x
{
x
:
x
is a perfect cube}
(viii)
{
x
:
x
+
5
=
8
}
\{x: x+5=8\}
{
x
:
x
+
5
=
8
}
(ix)
{
x
:
2
x
+
5
=
9
}
\{x: 2 x+5=9\}
{
x
:
2
x
+
5
=
9
}
(x)
{
x
:
x
≥
7
}
\{x: x \geq 7\}
{
x
:
x
≥
7
}
(xi)
{
x
:
x
∈
N
\{x: x \in \mathrm{N}
{
x
:
x
∈
N
and
2
x
+
1
>
10
}
2 x+1>10\}
2
x
+
1
>
10
}
Q4
EXERCISE 1.5
If
U
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
,
A
=
{
2
,
4
,
6
,
8
}
\mathrm{U}=\{1,2,3,4,5,6,7,8,9\}, \mathrm{A}=\{2,4,6,8\}
U
=
{
1
,
2
,
3
,
4
,
5
,
6
,
7
,
8
,
9
}
,
A
=
{
2
,
4
,
6
,
8
}
and
B
=
{
2
,
3
,
5
,
7
}
\mathrm{B}=\{2,3,5,7\}
B
=
{
2
,
3
,
5
,
7
}
. Verify that
(i)
(
A
∪
B
)
′
=
A
′
∩
B
′
(\mathrm{A} \cup \mathrm{B})^{\prime}=\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}
(
A
∪
B
)
′
=
A
′
∩
B
′
(ii)
(
A
∩
B
)
′
=
A
′
∪
B
′
(\mathrm{A} \cap \mathrm{B})^{\prime}=\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}
(
A
∩
B
)
′
=
A
′
∪
B
′
Q5
EXERCISE 1.5
Draw appropriate Venn diagram for each of the following :
(i)
(
A
∪
B
)
′
(\mathrm{A} \cup \mathrm{B})^{\prime}
(
A
∪
B
)
′
,
(ii)
A
′
∩
B
′
\mathrm{A}^{\prime} \cap \mathrm{B}^{\prime}
A
′
∩
B
′
,
(iii)
(
A
∩
B
)
′
(\mathrm{A} \cap \mathrm{B})^{\prime}
(
A
∩
B
)
′
,
(iv)
A
′
∪
B
′
\mathrm{A}^{\prime} \cup \mathrm{B}^{\prime}
A
′
∪
B
′
Q6
EXERCISE 1.5
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
∘
60^{\circ}
6
0
∘
, what is
A
′
\mathrm{A}^{\prime}
A
′
?
Q7
EXERCISE 1.5
Fill in the blanks to make each of the following a true statement :
(i)
A
∪
A
′
=
…
\mathrm{A} \cup \mathrm{A}^{\prime}=\dots
A
∪
A
′
=
…
(ii)
ϕ
′
∩
A
=
…
\phi^{\prime} \cap \mathrm{A}=\dots
ϕ
′
∩
A
=
…
(iii)
A
∩
A
′
=
…
\mathrm{A} \cap \mathrm{A}^{\prime}=\dots
A
∩
A
′
=
…
.
(iv)
U
′
∩
A
=
…
\mathrm{U}^{\prime} \cap \mathrm{A}=\dots
U
′
∩
A
=
…
Q1
Miscellaneous Exercise on Chapter 1
Decide, among the following sets, which sets are subsets of one and another:
A
=
{
x
:
x
∈
R
\mathrm{A}=\{x: x \in \mathbf{R}
A
=
{
x
:
x
∈
R
and
x
x
x
satisfy
x
2
−
8
x
+
12
=
0
}
x^{2}-8 x+12=0\}
x
2
−
8
x
+
12
=
0
}
,
B
=
{
2
,
4
,
6
}
,
C
=
{
2
,
4
,
6
,
8
,
…
}
,
D
=
{
6
}
\mathrm{B}=\{2,4,6\}, \quad \mathrm{C}=\{2,4,6,8, \dots\}, \mathrm{D}=\{6\}
B
=
{
2
,
4
,
6
}
,
C
=
{
2
,
4
,
6
,
8
,
…
}
,
D
=
{
6
}
.
Q2
Miscellaneous Exercise on Chapter 1
In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
(i)
If
x
∈
A
x \in \mathrm{A}
x
∈
A
and
A
∈
B
\mathrm{A} \in \mathrm{B}
A
∈
B
, then
x
∈
B
x \in \mathrm{B}
x
∈
B
(ii)
If
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
and
B
∈
C
\mathrm{B} \in \mathrm{C}
B
∈
C
, then
A
∈
C
\mathrm{A} \in \mathrm{C}
A
∈
C
(iii)
If
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
and
B
⊂
C
\mathrm{B} \subset \mathrm{C}
B
⊂
C
, then
A
⊂
C
\mathrm{A} \subset \mathrm{C}
A
⊂
C
(iv)
If
A
⊄
B
\mathrm{A} \not\subset \mathrm{B}
A
⊂
B
and
B
⊄
C
\mathrm{B} \not\subset \mathrm{C}
B
⊂
C
, then
A
⊄
C
\mathrm{A} \not\subset \mathrm{C}
A
⊂
C
(v)
If
x
∈
A
x \in \mathrm{A}
x
∈
A
and
A
⊄
B
\mathrm{A} \not\subset \mathrm{B}
A
⊂
B
, then
x
∈
B
x \in \mathrm{B}
x
∈
B
(vi)
If
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
and
x
∉
B
x \notin \mathrm{B}
x
∈
/
B
, then
x
∉
A
x \notin \mathrm{A}
x
∈
/
A
Q3
Miscellaneous Exercise on Chapter 1
Let
A
,
B
\mathrm{A}, \mathrm{B}
A
,
B
, and C be the sets such that
A
∪
B
=
A
∪
C
\mathrm{A} \cup \mathrm{B}=\mathrm{A} \cup \mathrm{C}
A
∪
B
=
A
∪
C
and
A
∩
B
=
A
∩
C
\mathrm{A} \cap \mathrm{B}=\mathrm{A} \cap \mathrm{C}
A
∩
B
=
A
∩
C
. Show that
B
=
C
\mathrm{B}=\mathrm{C}
B
=
C
.
Q4
Miscellaneous Exercise on Chapter 1
Show that the following four conditions are equivalent :
(i)
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
(ii)
A
−
B
=
ϕ
\mathrm{A}-\mathrm{B}=\phi
A
−
B
=
ϕ
(iii)
A
∪
B
=
B
\mathrm{A} \cup \mathrm{B}=\mathrm{B}
A
∪
B
=
B
(iv)
A
∩
B
=
A
\mathrm{A} \cap \mathrm{B}=\mathrm{A}
A
∩
B
=
A
Q5
Miscellaneous Exercise on Chapter 1
Show that if
A
⊂
B
\mathrm{A} \subset \mathrm{B}
A
⊂
B
, then
C
−
B
⊂
C
−
A
\mathrm{C}-\mathrm{B} \subset \mathrm{C}-\mathrm{A}
C
−
B
⊂
C
−
A
.
Q6
Miscellaneous Exercise on Chapter 1
Show that for any sets A and B,
A
=
(
A
∩
B
)
∪
(
A
−
B
)
A=(A \cap B) \cup (A-B)
A
=
(
A
∩
B
)
∪
(
A
−
B
)
and
A
∪
(
B
−
A
)
=
(
A
∪
B
)
A \cup (B-A)=(A \cup B)
A
∪
(
B
−
A
)
=
(
A
∪
B
)
Q7
Miscellaneous Exercise on Chapter 1
Using properties of sets, show that
(i)
A
∪
(
A
∩
B
)
=
A
\mathrm{A} \cup(\mathrm{A} \cap \mathrm{B})=\mathrm{A}
A
∪
(
A
∩
B
)
=
A
(ii)
A
∩
(
A
∪
B
)
=
A
\mathrm{A} \cap(\mathrm{A} \cup \mathrm{B})=\mathrm{A}
A
∩
(
A
∪
B
)
=
A
.
Q8
Miscellaneous Exercise on Chapter 1
Show that
A
∩
B
=
A
∩
C
\mathrm{A} \cap \mathrm{B}=\mathrm{A} \cap \mathrm{C}
A
∩
B
=
A
∩
C
need not imply
B
=
C
\mathrm{B}=\mathrm{C}
B
=
C
.
Q9
Miscellaneous Exercise on Chapter 1
Let A and B be sets. If
A
∩
X
=
B
∩
X
=
ϕ
\mathrm{A} \cap \mathrm{X}=\mathrm{B} \cap \mathrm{X}=\phi
A
∩
X
=
B
∩
X
=
ϕ
and
A
∪
X
=
B
∪
X
\mathrm{A} \cup \mathrm{X}=\mathrm{B} \cup \mathrm{X}
A
∪
X
=
B
∪
X
for some set X , show that
A
=
B
\mathrm{A}=\mathrm{B}
A
=
B
. (Hints
A
=
A
∩
(
A
∪
X
)
,
B
=
B
∩
(
B
∪
X
)
\mathrm{A}=\mathrm{A} \cap(\mathrm{A} \cup \mathrm{X}), \mathrm{B}=\mathrm{B} \cap(\mathrm{B} \cup \mathrm{X})
A
=
A
∩
(
A
∪
X
)
,
B
=
B
∩
(
B
∪
X
)
and use Distributive law)
Q10
Miscellaneous Exercise on Chapter 1
Find sets
A
,
B
\mathrm{A}, \mathrm{B}
A
,
B
and C such that
A
∩
B
,
B
∩
C
\mathrm{A} \cap \mathrm{B}, \mathrm{B} \cap \mathrm{C}
A
∩
B
,
B
∩
C
and
A
∩
C
\mathrm{A} \cap \mathrm{C}
A
∩
C
are non-empty sets and
A
∩
B
∩
C
=
ϕ
\mathrm{A} \cap \mathrm{B} \cap \mathrm{C}=\phi
A
∩
B
∩
C
=
ϕ
.
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