Practice set
1.3
Question
1:
If A = { a , b, c, d, e}, B = { c, d, e, f }, C = {b, d}, D = { a , e}
then which of the following statements are true and which are
false ? ( i ) C ⊆ B ( ii ) A
⊆ D ( iii ) D ⊆ B ( iv ) D⊆ A ( v ) B ⊆ A (vi) C ⊆ A
ANSWER:
We have, A = {a , b, c, d, e}, B = {c, d, e, f }, C =
{b, d}, D = {a, e}
(i) Since, b
∈ C but b ∉ B , b ∈ C but b ∉ B.
So, C ⊆ B is false.
(ii) Since, b
∈ A but b ∉ D ,b ∈ A but b ∉ D.
So, A ⊆ D is false.
(iii) Since, a
∈ D but a ∉ B ,a ∈ D but a ∉ B.
So, D ⊆ B is false.
(iv) Since, all
elements of set D is in set A.
So, D ⊆ A is true.
(v) Since, f
∈ B but f ∉ A ,f ∈ B but f ∉ A.
So, B ⊆ A is false.
(vi) Since, all
elements of set C is in set A.
So, C ⊆ A
Question 2:
Take the set of natural numbers from 1 to 20 as
universal set and show set X and Y using
Venn diagram.
( i ) X = { x
| x ∈
N, and 7 < x < 15}
( ii ) Y = { y
| y ∈ N, y is prime number from 1 to 20}
ANSWER:
We have,
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20};
X = {x | x
∈ N, and 7 < x
< 15} = {8, 9 ,10, 11, 12, 13, 14}; and
Y = {y | y
∈ N, y is prime number from 1 to 20} = {2, 3, 5, 7, 11, 13, 17, 19}
1 4 6 U
Y
8 9 2 3
10 11 7 5
X 12 13 17
14 19 15
20 16 18
Question
3:
U = {1, 2, 3, 7, 8, 9, 10, 11, 12}
P = {1, 3, 7, 10}
then
(i) show the sets U, P and P ¢ by Venn diagram
(ii) Verify (P ¢
) ¢ = P
ANSWER:
We have,
U = {1, 2,
3, 7, 8, 9, 10, 11, 12} and P = {1, 3, 7, 10}
P' = U − P = {2, 4,
8, 9, 11, 12}
(i)
(ii) (P')' = U − P' = {1,
3, 7, 10} = P
Question
4:
A = {1, 3, 2, 7} then write any three subsets of A.
ANSWER:
We have,
A =
{1, 3, 2, 7}
The three subsets of A are: {1},
{2} and {1, 2, 3}.
Disclaimer: There are 24 = 16 subsets possible
i.e. { }, {1}, {2}, {3}, {7}, {1, 2}, {1, 3}, {1, 7}, {2, 3}, {2, 7}, {3, 7}, {1, 2, 3}, {2, 3,
7}, {1, 3, 7}, {1, 2, 7} and {1, 2, 3,
7}.
Question 5:
( i) Write the subset relation between the sets.
P is the set of all residents in
Pune.
M is the set of all residents in Madhya
Pradesh.
I is the set of all residents in
Indore.
B is
the set of all residents in India.
H is the set of all residents in
Maharashtra.
( ii) Which set can be the universal set for above
sets ?
ANSWER:
(i) Sine
Pune is a city in Maharashtra, and Maharashtra is a state in India.
So, P ⊂
H ⊂ B
Similarly,
Indore is a city in Madhya
Pradesh, and Madhya Pradesh is a state in India.
So, I ⊂
M ⊂ B
(ii) The
set B can be the universal set for above sets.
Question 6:
Which set of numbers could be the
universal set for the sets given below?
(i)
A = set of multiples of 5, B = set of multiples of 7.
C = set of multiples of 12
(ii) P
= set of integers which are multiples of 4.
T = set of all even square numbers.1
ANSWER:
(i) We have,
A = set of multiples of 5 = {5,
10, 15, ...}; B = set of multiples of 7
= {7, 14, 21, ...}; and C = set of multiples of 12 = {12, 24, 36, ...}
The universal set for the sets A,
B and C can be set of natural numbers, i.e. N.
Disclaimer: The Universal set for
the sets A, B and C can be set of whole numbers, integers, rational number of
real numbers and so on.
(ii) We have,
P = set of integers which are
multiples of 4 = {4, 8, 12, ...}; and
T =
set of all even square numbers = {4, 16, 36, 64, 100, ...}
So, P can be the Universal set
for the sets P and Q.
Disclaimer: Here also, the Universal set can be vary from
the above.
Question 7:
Let all the students of a class is an Universal set. Let
set A be the students who secure 50% or more marks in Maths. Then write
the complement of set A.1
ANSWER:
We have,
U =
set of all the students of a class and A = set of the students who secure 50%
or more marks in Maths