SETS: PROBLEM SET
Question 1:
Choose
the correct alternative answer for each of the following questions.
(i)
If M = {1, 3, 5}, N = {2, 4, 6}, then M∩N = ?
(A) {1, 2, 3, 4, 5, 6} (B) {1, 3, 5} (C) Ï• (D) {2, 4, 6}
(ii)
P = { x | x is an odd natural number, 1 < x ≤5}
How to write this set in roster form?
(A) {1, 3, 5} (B) {1, 2, 3, 4, 5} (C) {1, 3} (D) {3, 5}
(iii) P = {1, 2, ........., 10}, What type of set P is ?
(A) Null set (B) Infinite set (C) Finite set (D) None of
these
(iv)
M∪N = {1, 2, 3, 4, 5, 6} and M = {1, 2, 4} then
which of the following represent set N ?
(A) {1, 2, 3} (B) {3, 4, 5, 6} (C) {2, 5, 6} (D) {4, 5, 6}
(v)
If P⊆ M, then Which of the following set
represent P ∩ (P ∪M) ?
(A)
P (B) M (C) P∪M (D) P' ∩ M
(vi) Which of the following sets are empty sets ?
(A)
set of intersecting points of parallel lines
(B) set of even prime numbers.
(C)
Month of an english calendar having less than 30 days.
(D)
P = { x | x ∈ I, −1 < x <
1}
ANSWER:
(i)
We have,
M = {1, 3, 5}, N = {2, 4, 6}
M∩N =Ï• = Empty set
So, the correct option is (C).
(ii)
Since, P = {x | x is an odd natural number, 1
< x ≤5} = {3, 5}
So,
the correct option is (D).
(iii) Since, the elements of set P = {1, 2, ..., 10} is finite.
So, set P is a finite set.
Hence, the correct option is (C).
(iv) We
have, M ∪ N = {1, 2, 3, 4, 5, 6} and M = {1, 2,
4}
Since, {1, 2, 3} ∪ {1, 2, 4} = {1, 2, 3, 4} ≠ M ∪ N = {1, 2, 3, 4, 5,
6};
{3,
4, 5, 6} ∪ {1, 2, 4} = {1, 2, 3, 4, 5, 6}
= M ∪ N = {1, 2, 3, 4, 5,6};
{2, 5, 6} ∪ {1, 2, 4} = {1, 2, 4, 5, 6} ≠ M ∪ N = {1, 2, 3, 4, 5,
6}; and
{4, 5, 6} ∪ {1, 2, 4} = {1, 2, 4, 5, 6} ≠ M ∪ N = {1, 2, 3, 4, 5,
6}
So,
the correct option is (B).
(v) We
have, P⊆ M,
Now, P ∩ (P ∪ M) = P ∩ M = M
(Since, P ∪ M = M; P⊆ M)
So, the correct option is (B).
(vi) Since,
the set of intersecting points of parallel lines = {};
the set of even prime numbers= {2};
the
Month of an english calendar having less than 30 days = {February}; and
P = {x | x ∈ I, −1 < x <
1} = {0}
So, the correct option is (A).
Question 2:
Find
the correct option for the given question.
(i) Which of the following collections is a set ?
(A)
Colours of the rainbow (B) Tall trees in the school campus. (C) Rich
people in the village (D) Easy examples in the book
(ii) Which of the following set represent N ∩ W?
(A)
{1, 2, 3, .....} (B) {0, 1, 2, 3, ....} (C) {0}
(D) { }
(iii) P = { x | x is
a letter of the word ' indian'} then which one of the following is set P in
listing form ?
(A)
{i, n, d} (B) {i, n, d, a} (C) {i, n, d, i, a} (D) {n, d, a}
(iv) If T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8} then T∪M = ?
(A)
{1, 2, 3, 4, 5, 7} (B) {1, 2, 3, 7, 8}
(C) {1, 2, 3, 4, 5, 7, 8} (D) {3, 4}
ANSWER:
(i)
(A) Since, the colours of the rainbow are well defined such as Violet, Indigo,
Blue, Green, Yellow, Orange and Red.So,
the collection of colours of the rainbow is a set.
(B) Since, the tall trees in the school campus are not well defined.
So, the collection of the tall trees in the school campus is not a set.
(C) Since, the rich people in the village is not well defined.So,
the colection of the rich people in the village is not a set.
(D) Since, the easy examples in the book is not well defined.
So, the collection of easy examples in the book is not a set.
(ii) Since, N ∩∩ W = {1, 2, 3, 4, ...}
So, the correct option is (A).
(iii)
Since, P = { x | x is a letter of the word
'indian'}
So, P = {i, n, d, i, a, n} = {i, n, d, a}
Hence,
the correct option is (B).
(iv) Since,
T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8}
So, T∪∪M = {1, 2, 3, 4, 5, 7, 8}
Hence, the correct option is (C).
Question 3:
Out
of 100 persons in a group, 72 persons speak English and 43 persons speak
French. Each one out of 100 persons speak at least one language. Then how many
speak only English ? How many speak only French ? How many of them speak
English and French both ?
ANSWER:
Let A be the set of persons speaking English and B be the
set of persons speaking French.
So, n (A) = 72; n (B) = 43; n (A ∪ B) = 100 n (A ∪ B) = 100
Now,
n (A) + n (B) = n (A ∪ B) + n (A ∩ B)
⇒ n (A ∩ B) = 72 +43 −100⇒ n (A ∩ B) = 15 So, the number of person who speak French and English both is 15.⇒ n (A ∩ B) = 72 +43 -100⇒ n (A ∩ B) = 15 So, the number of person who speak French and English both is 15.
Also,
n (A) = n (A − B) + n (A ∩ B)n (A) = n (A - B) + n (A ∩ B)
⇒ n (A − B) = 72 − 15⇒ n (A − B) = 57 So, the number of person who speak only English is 57.⇒ n (A - B) = 72 - 15⇒ n (A - B) = 57 So, the number of person who speak only English is 57.
And,
n (B) = n ( B − A) + n (A ∩ B)⇒ n (B − A) = 43 −15⇒ n (B − A) = 28
So, the number of person who speak only French is 28.
n(B) = n (B -A) + n (A ∩ B)⇒ n (B - A) = 43 - 15⇒ n (B - A) = 28
So, the number of person who speak only French is 28.
Question 4:
70
trees were planted by Parth and 90 trees were planted by Pradnya on the
occasion of Tree Plantation Week. Out of these; 25 trees were planted by both
of them together. How many trees were planted by Parth or Pradnya ?
ANSWER:
Let A be the set of tress planted by Parth and B be the
set of trees planted by Pradnya.
So, n (A) = 70; n (B) = 90; n (A ∩ B) = 25
Now,
n (A) + n (B) = n (A∪B) + n (A∩B)
n (A∪B) = 70 + 90 - 25
(A∪B) = 135
Hence, the number of trees planted by Parth or Pradnya is 135.
Question 5:
If n (A) = 20, n (B) = 28 and n (A∪B) = 36 then n (A ∩ B) = ?
ANSWER:
We have,
n (A) = 20, n (B) = 28 and n (A∪B) = 36
Since, n (A ∩ B ) = n (A) + n (B) − n (A ∪ B) = 20 + 28 - 36
∴ n (A ∩ B ) = 12
Question 6:
In
a class, 8 students out of 28 have a dog as their pet animal at home, 6
students have a cat as their pet animal. 10 students have dog and cat both,
then how many students do not have a dog or cat as their pet animal at home ?
ANSWER:
We have,
Total number of students = 28;
Students have a dog as their pet = 8;
Students have a cat as their pet = 6; and
Students have cat and dog both = 10
Solving using venn diagram, we get:
So,
the number of students that do not have a dog or a cat as their pet is 4.
Question 7:
Represent
the union of two sets by Venn diagram for each of the following.
(i) A ={3, 4, 5, 7} B ={1, 4, 8}
(ii)
P = {a, b, c, e, f}
Q ={l, m, n , e, b}
(iii)
X = { x | x is a prime number between 80 and
100}
Y = { y | y is an odd
number between 90 and 100 }
ANSWER:
(i) A ={3, 4, 5, 7} B
={1, 4, 8}
(ii) P = {a, b, c, e, f}
Q ={l, m, n , e, b}
(iii)
X = { x | x is a prime number between 80 and
100} = {83, 89, 97};
Y
= { y | y is an odd number between 90 and 100
} = {91, 93, 95, 97, 99}
Question 8:
Write
the subset relations between the following sets..
X = set of all quadrilaterals. Y = set of all rhombuses.
S =
set of all squares. T = set of all parallelograms.
V =
set of all rectangles.
ANSWER:
Since,
all squares are rectangle, all rectangles are parallelogram, all parallelograms
are quadrilateral; and all squares are rhombus, all rhombus are parallelogram,
all parallelograms are quadrilateral.
So, the subset relations are:
S < V < T < X and S < Y < T < X
Question 9:
If M is any set, then write M ∪ Ï• and M∩ Ï•.
ANSWER:
If M is any set, then
M ∪ Ï•=M
and,
M ∩ Ï•=Ï•
Question 10:
Observe the Venn diagram and write the given sets
U, A, B, A ∪ B, A ∩ B
ANSWER:
(i) U = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13}
(ii) A = {1, 2, 3, 5, 7}
(iii) B = {1, 5, 8, 9, 10}
(iv) A ∪ B = {1, 2, 3, 5, 7, 8, 9, 10}
(v) A∩B={1, 5}
Question 11:
If
n (A) = 7, n (B) = 13, n (A ∩ B )= 4, then n
(A ∪ B)=?
ANSWER:
We have,
n (A) = 7, n (B) = 13 and n (A ∩ B) = 4
Since, n (A ∪ B) = n (A) + n (B) − n (A ∩ B) = 7 + 13 − 4
∴ n (A ∪ B) = 16