Operation of Sets
Exercise 1
1. In the given Venn diagram, F represents the set of students who like football and V represents the set of students who like volleyball. Write the elements of the following sets by listing method:
1. Write the elements of the following sets by listing method: (a) F (Football set)F = {Chhiring, Dorje, Hari, Maya, Asha, Ganesh}
(b) V (Volleyball set)V = {Batuli, Dolma, Suntati, Maya, Asha, Ganesh}
(c) F ∪ V (Union of F and V: Students who like at least one sport)F ∪ V = {Chhiring, Dorje, Hari, Maya, Asha, Ganesh, Batuli, Dolma, Suntati}
(d) F ∩ V (Intersection of F and V: Students who like both football and volleyball)F ∩ V = {Maya, Asha, Ganesh}
(e) U (Universal set: All students mentioned in the diagram)U = {Chhiring, Dorje, Hari, Maya, Asha, Ganesh, Batuli, Dolma, Suntati, Rambahadur, Harkabahadur}
Mathematics Questions
Mathematics Questions
2. If \( U = \{ a, b, c, d, e, f, g, h, i, j, k \} \), \( A = \{ a, c, e, f, g, i, k \} \) and \( B = \{ b, d, i, j, k, h \} \), find the following sets and also present in a separate Venn diagram:
a. \( (A \cap B) \)
b. \( (B \cup A) \)
c. \( A - B \)
d. \( B - A \)
Solution:
a. \( (A \cap B) \)
\( A \cap B = \{ a, c, e, f, g, i, k \} \cap \{ b, d, i, j, k, h \} \)
= { i, k }
b. \( (A \cup B) \)
\( A \cup B = \{ a, c, e, f, g, i, k \} \cup \{ b, d, i, j, k, h \} \)
= { a, b, c, d, e, f, g, h, i, j, k }
= U
c. \( A - B \)
\( A - B = \{ a, c, e, f, g, i, k \} - \{ b, d, i, j, k, h \} \)
= { a, c, e, f, g }
d. \( B - A \)
\( B - A = \{ b, d, i, j, k, h \} - \{ a, c, e, f, g, i, k \} \)
= { b, d, h, j }
3. If \( U = \{ x: x \text{ is a whole number from 1 to 30} \} \), \( A = \{ x: x \text{ is a multiple of 3 from 1 to 30} \} \), \( B = \{ x: x \text{ is a multiple of 4 from 1 to 30} \} \) and \( C = \{ x: x \text{ is a multiple of 5 from 1 to 30} \} \), write the following sets by listing method and present them in a Venn diagram:
a. \( A - B \)
b. \( B - A \)
c. \( A - C \)
d. \( B - C \)
e. \( A \cup B \)
f. \( A \cup B \cup C \)
g. \( A \cap B \cap C \)
h. \( \overline{B \cup C} \)
Solution:
Given Sets:
U = { x: x is a whole number from 1 to 30 }
= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30 }
A = { x: x is a multiple of 3 from 1 to 30 }
= { 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 }
B = { x: x is a multiple of 4 from 1 to 30 }
= { 4, 8, 12, 16, 20, 24, 28 }
C = { x: x is a multiple of 5 from 1 to 30 }
= { 5, 10, 15, 20, 25, 30 }
a. \( A - B \)
\( A - B = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \} - \{ 4, 8, 12, 16, 20, 24, 28 \} \)
= { 3, 6, 9, 15, 18, 21, 27, 30 }
b. \( B - A \)
\( B - A = \{ 4, 8, 12, 16, 20, 24, 28 \} - \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \} \)
= { 4, 8, 16, 20, 28 }
c. \( A - C \)
\( A - C = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \} - \{ 5, 10, 15, 20, 25, 30 \} \)
= { 3, 6, 9, 12, 18, 21, 24, 27 }
d. \( B - C \)
\( B - C = \{ 4, 8, 12, 16, 20, 24, 28 \} - \{ 5, 10, 15, 20, 25, 30 \} \)
= { 4, 8, 12, 16, 24, 28 }
e. \( A \cup B \)
\( A \cup B = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \} \cup \{ 4, 8, 12, 16, 20, 24, 28 \} \)
= { 3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30 }
f. \( A \cup B \cup C \)
\( A \cup B \cup C = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \} \cup \{ 4, 8, 12, 16, 20, 24, 28 \} \cup \{ 5, 10, 15, 20, 25, 30 \} \)
= { 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30 }
g. \( A \cap B \cap C \)
\( A \cap B \cap C = \{ 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \} \cap \{ 4, 8, 12, 16, 20, 24, 28 \} \cap \{ 5, 10, 15, 20, 25, 30 \} \)
= { 12, 30 }
h. \( \overline{B \cup C} \)
\( B \cup C = \{ 4, 5, 8, 10, 12, 15, 16, 20, 24, 25, 28, 30 \} \)
\( \overline{B \cup C} = U - (B \cup C) \)
= { 1, 2, 3, 6, 7, 9, 11, 13, 14, 17, 18, 19, 21, 22, 23, 26, 27, 29 }
4. If \( U = \{ a, b, c, d, e, f, g, h, i, j, k \} \), \( A = \{ a, c, d, f \} \), \( B = \{ g, h, i \} \), then:
(a) Construct the following sets:
i. \( \overline{A} \)
ii. \( \overline{B} \)
iii. \( \overline{A} \cup \overline{B} \)
iv. \( \overline{A} \cap \overline{B} \)
v. \( \overline{A \cup B} \)
vi. \( \overline{A \cap B} \)
(b) Which of the sets in question A are equal? Find.
(a)
Solution:
U = { a, b, c, d, e, f, g, h, i, j, k }
A = { a, c, d, f }
B = { g, h, i }
i. Ā
= U - A
= { a, b, c, d, e, f, g, h, i, j, k } - { a, c, d, f }
= { b, e, g, h, i, j, k }
ii. B̅
= U - B
= { a, b, c, d, e, f, g, h, i, j, k } - { g, h, i }
= { a, b, c, d, e, f, j, k }
iii. Ä€ ∪ B̅
= { b, e, g, h, i, j, k } ∪ { a, b, c, d, e, f, j, k }
= { a, b, c, d, e, f, g, h, i, j, k }
iv. Ä€ ∩ B̅
= { b, e, g, h, i, j, k } ∩ { a, b, c, d, e, f, j, k }
= { b, e, j, k }
v. Ä€ ∪ B̅
= U - (A ∪ B)
= { a, c, d, f, g, h, i }
= { b, e, j, k }
vi. Ä€ ∩ B̅
= U - (A ∩ B)
= U - ∅
= U
= { a, b, c, d, e, f, g, h, i, j, k }
(b)
Ä€ ∪ B̅ = U = Ä€ ∩ B̅
Ä€ ∩ B̅ = Ä€ ∪ B̅ = { b, e, j, k }
5. If \( U = \{ \text{set of actual numbers from 1 to 12} \} \), \( E = \{ \text{set of even numbers from 1 to 12} \} \), \( O = \{ \text{set of odd numbers from 1 to 12} \} \), and \( P = \{ \text{set of prime numbers from 1 to 12} \} \), find the following sets and also present them in a Venn diagram:
a. \( \overline{E} \)
b. \( \overline{O} \)
c. \( \overline{P} \)
d. \( E \cup P \)
e. \( P \cap Q \)
f. \( P - O \)
g. \( P \)
h. \( \overline {E \cup O \cup P}\)
i. \( \overline {(E \cap O \cap P}) \)
j. \( \overline {(E \cup P)} - (P \cap O) \)
k. \( \overline{P} \cup (E) \cap O \)
Solution:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }
E = { 2, 4, 6, 8, 10, 12 } (Set of even numbers)
O = { 1, 3, 5, 7, 9, 11 } (Set of odd numbers)
P = { 2, 3, 5, 7, 11 } (Set of prime numbers)
a. \( \overline{E} \)
= U - E
= { 1, 3, 5, 7, 9, 11 }
b. \( \overline{O} \)
= U - O
= { 2, 4, 6, 8, 10, 12 }
c. \( \overline{P} \)
= U - P
= { 1, 4, 6, 8, 9, 10, 12 }
d. \( E \cup P \)
= { 2, 4, 6, 8, 10, 12 } ∪ { 2, 3, 5, 7, 11 }
= { 2, 3, 4, 5, 6, 7, 8, 10, 11, 12 }
e. \( P \cap O \)
= { 2, 3, 5, 7, 11 } ∩ { 1, 3, 5, 7, 9, 11 }
= { 3, 5, 7, 11 }
f. \( P - O \)
= { 2, 3, 5, 7, 11 } - { 1, 3, 5, 7, 9, 11 }
= { 2 }
g. \( P \)
= { 2, 3, 5, 7, 11 }
h. \( \overline {E \cup O \cup P} \)
= U - (E ∪ O ∪ P)
= U - { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }
= { } (Empty set)
i. \( \overline {(E \cap O \cap P)} \)
= U - (E ∩ O ∩ P)
= U - { } (Since there is no common element in E, O, and P)
= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }
j. \( \overline {(E \cup P)} - (P \cap O) \)
= (U - (E ∪ P)) - (P ∩ O)
= { 1, 9 } - { 3, 5, 7, 11 }
= { 1, 9 }
k. \( \overline{P} \cup (E \cap O) \)
= { 1, 4, 6, 8, 9, 10, 12 } ∪ { } (Since E and O have no common elements)
= { 1, 4, 6, 8, 9, 10, 12 }
Solution:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }
E = { 2, 4, 6, 8, 10, 12 } (Set of even numbers)
O = { 1, 3, 5, 7, 9, 11 } (Set of odd numbers)
P = { 2, 3, 5, 7, 11 } (Set of prime numbers)
a. \( \overline{E} \)
= U - E
= { 1, 3, 5, 7, 9, 11 }
b. \( \overline{O} \)
= U - O
= { 2, 4, 6, 8, 10, 12 }
c. \( \overline{P} \)
= U - P
= { 1, 4, 6, 8, 9, 10, 12 }
d. \( E \cup P \)
= { 2, 4, 6, 8, 10, 12 } ∪ { 2, 3, 5, 7, 11 }
= { 2, 3, 4, 5, 6, 7, 8, 10, 11, 12 }
e. \( P \cap O \)
= { 2, 3, 5, 7, 11 } ∩ { 1, 3, 5, 7, 9, 11 }
= { 3, 5, 7, 11 }
f. \( P - O \)
= { 2, 3, 5, 7, 11 } - { 1, 3, 5, 7, 9, 11 }
= { 2 }
g. \( P \)
= { 2, 3, 5, 7, 11 }
h. \( \overline {E \cup O \cup P} \)
= U - (E ∪ O ∪ P)
= U - { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }
= { } (Empty set)
i. \( \overline {(E \cap O \cap P)} \)
= U - (E ∩ O ∩ P)
= U - { } (Since there is no common element in E, O, and P)
= { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 }
j. \( \overline {(E \cup P)} - (P \cap O) \)
= (U - (E ∪ P)) - (P ∩ O)
= { 1, 9 } - { 3, 5, 7, 11 }
= { 1, 9 }
k. \( \overline{P} \cup (E \cap O) \)
= { 1, 4, 6, 8, 9, 10, 12 } ∪ { } (Since E and O have no common elements)
= { 1, 4, 6, 8, 9, 10, 12 }
6. If \( U = \{ m, n, o, p, q, r, s, t, u, v \} \), \( A = \{ q, r, s, t, u, v \} \), \( B = \{ n, o, p, q, r \} \), and \( C = \{ m, u, s, t, q, r \} \), find the sets of the following relations. Also, present them by drawing separate Venn diagrams:
a. \( A \cap B \)
b. \( (A \cup B) \cap C \)
c. \( (A \cup B \cup C) \)
d. \( A \cap B \cap C \)
e. \( A - B \)
f. \( \overline {(A \cup B \cup C)} \)
g. \(\overline {A \cap B }\)
h. \(\overline {A} \)
i. \((A \cap C) \cup B \)
Solution:
U = { m, n, o, p, q, r, s, t, u, v }
A = { q, r, s, t, u, v }
B = { n, o, p, q, r }
C = { m, u, s, t, q, r }
a. \( A \cap B \)
= { q, r, s, t, u, v } ∩ { n, o, p, q, r }
= { q, r }
b. \( (A \cup B) \cap C \)
= ({ q, r, s, t, u, v } ∪ { n, o, p, q, r }) ∩ { m, u, s, t, q, r }
= { n, o, p, q, r, s, t, u, v } ∩ { m, u, s, t, q, r }
= { q, r, s, t, u }
c. \( (A \cup B \cup C) \)
= { q, r, s, t, u, v } ∪ { n, o, p, q, r } ∪ { m, u, s, t, q, r }
= { m, n, o, p, q, r, s, t, u, v }
= U
d. \( A \cap B \cap C \)
= { q, r, s, t, u, v } ∩ { n, o, p, q, r } ∩ { m, u, s, t, q, r }
= { q, r }
e. \( A - B \)
= { q, r, s, t, u, v } - { n, o, p, q, r }
= { s, t, u, v }
f. \( \overline {(A \cup B \cup C)} \)
= U - (A ∪ B ∪ C)
= { m, n, o, p, q, r, s, t, u, v } - { m, n, o, p, q, r, s, t, u, v }
= { } (Empty set)
g. \( \overline {A \cap B} \)
= U - (A ∩ B)
= { m, n, o, p, q, r, s, t, u, v } - { q, r }
= { m, n, o, p, s, t, u, v }
h. \( \overline {A} \)
= U - A
= { m, n, o, p, q, r, s, t, u, v } - { q, r, s, t, u, v }
= { m, n, o, p }
i. \( (A \cap C) \cup B \)
= ({ q, r, s, t, u, v } ∩ { m, u, s, t, q, r }) ∪ { n, o, p, q, r }
= { q, r, s, t, u } ∪ { n, o, p, q, r }
= { n, o, p, q, r, s, t, u }
8. If P and Q are the intersecting subsets of an universal set U, show the following sets by drawing Venn diagram:(a) P - Q (b) Q - P (c) (P - Q) ∪ P (d) P ∩ (Q - P)
8. If P and Q are the intersecting subsets of an universal set U, show the following sets by drawing Venn diagram:(a) P - Q (b) Q - P (c) (P - Q) ∪ P (d) P ∩ (Q - P)
(a) \( P - Q \)
= { Elements in P but not in Q }
(b) \( Q - P \)
= { Elements in Q but not in P }
(c) \( (P - Q) \cup P \)
= { Elements in P - Q } ∪ { Elements in P }
= P (since (P - Q) is already a subset of P)
(d) \( P \cap (Q - P) \)
= { Elements that are in P and also in Q but not in P }
= ∅ (Empty set, since no elements can be in P and also in Q but not in P)
9. Write an universal set U and two subsets X and Y. After that, write the elements of following sets by listing method:(a) (X̅ ∪ Y) (b) (X ∩ Y̅) (c) X̅ (d) X̅ ∩ Y̅
Let \( U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} \)
\( X = \{ 2, 4, 6, 8, 10 \} \) (even numbers)
\( Y = \{ 1, 2, 3, 4, 5 \} \) (first five natural numbers)
(a) \( X̅ \cup Y \)
= (U - X) ∪ Y
= { 1, 3, 5, 7, 9 } ∪ { 1, 2, 3, 4, 5 }
= { 1, 2, 3, 4, 5, 7, 9 }
(b) \( X \cap Y̅ \)
= X ∩ (U - Y)
= { 2, 4, 6, 8, 10 } ∩ { 6, 7, 8, 9, 10 }
= { 6, 8, 10 }
(c) \( X̅ \)
= U - X
= { 1, 3, 5, 7, 9 }
(d) \( X̅ \cap Y̅ \)
= (U - X) ∩ (U - Y)
= { 1, 3, 5, 7, 9 } ∩ { 6, 7, 8, 9, 10 }
= { 7, 9 }
Exercise 1.2
1.(a) Observe the given Venn diagram and find the cardinality of the following sets:
Solution:
(a) n (S)S = {1, 2, 3, 5, 6}
∴ n (S) = 5
(b) n (T)T = {1, 2, 4, 7, 8, 9}
∴ n (T) = 6
(c) n(U)U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
∴ n (U) = 12
(d) n (S ∩ T)S ∩ T = {1, 2}
∴ n(S ∩ T) = 2
(e) n (R ∪ T)R ∪ T = {1, 2, 3, 4, 7, 8, 9, 10}
∴ n (R ∪ T) = 8
(f) n (S ∩ R ∩ T)S ∩ R ∩ T = {1}
∴ n(S ∩ R ∩ T) = 1
(g) n (S ∪ R ∪ T)S ∪ R ∪ T = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
∴ n (S ∪ R ∪ T) = 10
(h) n(S ∪ R ∪ T)(S ∪ R ∪ T) = {11, 12}
∴ n (S ∪ R ∪ T) = 2
(i) n₀ (S)Elements not in S = {4, 7, 8, 9, 10, 11, 12}
∴ n₀(S) = 7
(j) n₀ (S ∩ R)Elements not in S ∩ R = {2, 4, 5, 6, 7, 8, 9, 10, 11, 12}
∴ n₀ (S ∩ R) = 10
(k) n(R)Elements not in R = {2, 5, 6, 7, 8, 9, 11, 12}
∴ n(R) = 8
(b) If A and B are any two disjoint sets, n (A) = 30, n (B) = 35, find the value of n (A ∪ B).
Solution:
Since A and B are disjoint sets, their union has the sum of their cardinalities.
n (A ∪ B) = n (A) + n (B)
= 30 + 35
= 65
2.(a) From the given Venn diagram, find the cardinality of the following sets:
(a) n(A)A = { a, b, c, d, e, f, g, k, l }
∴ n (A) = 9
(b) n (B)B = { a, b, c, h, i, j, m }
∴ n (B) = 7
(c) n (A ∪ B)A ∪ B = { a, b, c, d, e, f, g, h, i, j, k, l, m }
∴ n (A ∪ B) = 13
(d) n (A ∩ B)A ∩ B = { a, b, c }
∴ n (A ∩ B) = 3
(e) n₀ (A)Elements outside A in U = U - A = { h, i, j, m }
∴ n₀ (A) = 4
(f) n₀ (B)Elements outside B in U = U - B = { d, e, f, g, k, l }
∴ n₀ (B) = 6
(b) If U = {a, b, c, d, e, f, g}, A = {c, d, e, f}, B = {a, b, e, f}, C = {d, e, f, g}, find the cardinality of the following sets:
Solution:
(a) n (A - B)
A - B = { c, d } (elements in A but not in B)
∴ n (A - B) = 2
(b) n (B - C)
B - C = { a, b } (elements in B but not in C)
∴ n (B - C) = 2
(c) n (A - C)
A - C = { c } (elements in A but not in C)
∴ n (A - C) = 1
(d) n(A)
U - A = { a, b, g } (complement of A in U)
∴ n( A ) = 3
(e) n(A ∪ B)
A ∪ B = { a, b, c, d, e, f }
U - (A ∪ B) = { g }
∴ n ( A ∪ B ) = 1
(f) n {(A ∪ B) - (A ∩ B)}
A ∪ B = { a, b, c, d, e, f }
A ∩ B = { e, f }
(A ∪ B) - (A ∩ B) = { a, b, c, d }
∴ n {(A ∪ B) - (A ∩ B)} = 4
(g) n {(A - B) ∪ (B - A)}
A - B = { c, d }
B - A = { a, b }
(A - B) ∪ (B - A) = { a, b, c, d }
∴ n {(A - B) ∪ (B - A)} = 4
3.(a) If U = {set of natural numbers less than 20}, A = {set of even numbers less than 20}, B = {set of prime numbers less than 20} and C = {set of square numbers less than 20}, find the cardinality of the following sets:
Solution:
(a) n (U)
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
∴ n (U) = 19
(b) n (C)
C = {1, 4, 9, 16}
∴ n (C) = 4
(c) n(A ∩ B)
A = {2, 4, 6, 8, 10, 12, 14, 16, 18}
B = {2, 3, 5, 7, 11, 13, 17, 19}
A ∩ B = {2}
∴ n (A ∩ B) = 1
(d) n (B - C)
B = {2, 3, 5, 7, 11, 13, 17, 19}
C = {1, 4, 9, 16}
B - C = {2, 3, 5, 7, 11, 13, 17, 19}
∴ n (B - C) = 8
(e) n (A)
A = {2, 4, 6, 8, 10, 12, 14, 16, 18}
∴ n (A) = 9
(f) n (A ∪ C)
A = {2, 4, 6, 8, 10, 12, 14, 16, 18}
C = {1, 4, 9, 16}
A ∪ C = {1, 2, 4, 6, 8, 9, 10, 12, 14, 16, 18}
∴ n (A ∪ C) = 11
(g) n (A ∩ B)
A ∩ B = {2}
∴ n (A ∩ B) = 1
(b) If U = {x: x is a natural number less than 20}, A = {y: y is a prime number}, B = {z: z is a factor of 18}, C = {p: p is a multiple of 3 less than 20}, present the following sets by drawing separate Venn diagrams and find their cardinality:
Solution:
(a) n (A ∪ B)
A = {2, 3, 5, 7, 11, 13, 17, 19}
B = {1, 2, 3, 6, 9, 18}
A ∪ B = {1, 2, 3, 5, 6, 7, 9, 11, 13, 17, 18, 19}
∴ n (A ∪ B) = 12
(b) n (B ∪ C)
B = {1, 2, 3, 6, 9, 18}
C = {3, 6, 9, 12, 15, 18}
B ∪ C = {1, 2, 3, 6, 9, 12, 15, 18}
∴ n (B ∪ C) = 8
(c) n (A ∪ B ∪ C)
A = {2, 3, 5, 7, 11, 13, 17, 19}
B = {1, 2, 3, 6, 9, 18}
C = {3, 6, 9, 12, 15, 18}
A ∪ B ∪ C = {1, 2, 3, 5, 6, 7, 9, 11, 12, 13, 15, 17, 18, 19}
∴ n (A ∪ B ∪ C) = 14
(d) n (A ∩ B ∩ C)
A ∩ B ∩ C = {3}
∴ n (A ∩ B ∩ C) = 1
(e) n₀ (A)
A = {2, 3, 5, 7, 11, 13, 17, 19}
U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}
Complement of A in U, U – A = {1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}
∴ n₀ (A) = 11
(f) n₀ (A - B)
A = {2, 3, 5, 7, 11, 13, 17, 19}
B = {1, 2, 3, 6, 9, 18}
A - B = {5, 7, 11, 13, 17, 19}
Complement of (A - B) in U, U – (A - B) = {1, 2, 3, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18}
∴ n₀ (A - B) = 13
EXERCISE 1.1
General section
1. Let P and Q are the subsets of a universal set U. Write the set operations defined by the following set-builder forms:
a) { x: x ∈ P or x ∈ Q }
b) { x: x ∈ P and x ∈ Q }
c) { x: x ∈ P but x ∉ Q }
d) { x: x ∈ Q but x ∉ P }
e) { x: x ∈ U but x ∉ P }
f) { x: x ∈ U but x ∉ Q }
g) { x: x ∈ U but x ∈ P or x ∈ Q }
h) { x: x ∈ U but x ∉ P and x ∉ Q }
i) { x: x ∈ U but x ∈ P − Q }
Solution:
a) { x: x ∈ P or x ∈ Q } = P ∪ Q
b) { x: x ∈ P and x ∈ Q } = P ∩ Q
c) { x: x ∈ P but x ∉ Q } = P − Q
d) { x: x ∈ Q but x ∉ P } = Q − P
e) { x: x ∈ U but x ∉ P } = P
f) { x: x ∈ U but x ∉ Q } = Q
g) { x: x ∈ U but x ∉ P or x ∉ Q } = P ∪ Q
h) { x: x ∈ U but x ∉ P and x ∉ Q } = P ∩ Q
i) { x: x ∈ U but x ∈ P − Q } = P - Q
2. Write the set operations represented by shaded regions shown in the following Venn diagrams.
a) The shaded region represents A ∪ B
b) The shaded region represents A ∪ B
c) The shaded region represents A ∩ B
d) The shaded region represents A ∩ B
e) The shaded region represents A − B
f) The shaded region represents Y − X
g) The shaded region represents A ∪ B ∪ C
h) The shaded region represents P ∩ Q ∩ R
3. (a) A and B are subsets of the universal set U. From the given diagrams, list the elements of the following set operations:
(i) A ∪ B and A ∪ B
(ii) A ∩ B and A ∩ B
(iii) A − B and A - B
(iv) B − A and B − A
Solution:(i) A ∪ B = {1, 2, 3, 5, 6, 7, 9} and A ∪ B = {4, 8, 10}
(ii) A ∩ B = {2, 3, 5, 6, 7, 9} and A ∩ B = {1}
(iii) A − B = {1, 4} and A - B = {2, 3, 5, 6, 7, 9}
(iv) B − A = {8, 10} and B − A = {2, 3, 5, 6, 7, 9}
(b) P, Q, and R are subsets of the universal set U. List the elements of the following set operations from the given diagram:
(i) P ∪ Q ∪ R
(ii) P ∩ Q ∩ R
(iii) P ∪ Q ∪ R
(iv) P ∩ Q ∩ R
(v) (P ∪ Q) ∩ R
(vi) (P ∩ Q) ∪ R
Solution:(i) P ∪ Q ∪ R = {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15}
(ii) P ∩ Q ∩ R = {6}
(iii) P ∪ Q ∪ R = {7, 11, 13, 14}
(iv) P ∩ Q ∩ R = {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15}
(v) (P ∪ Q) ∩ R = {1, 2, 3, 4, 6, 8, 10, 12} ∩ {3, 6, 9, 12, 15} = {3, 6}
(vi) (P ∩ Q) ∪ R = {2, 4, 6} ∪ {3, 6, 9, 12, 15} = {2, 3, 4, 6, 9, 12, 15}
upto 34.
a) If A = { n, e, p, a, l } and B = { b, h, u, t, a, n }, find
(i) A ∪ B = { a, b, e, h, l, n, p, t, u }
(ii) A ∩ B = { a, n }
(iii) A - B = { e, p, l }
(iv) B - A = { b, h, t, u }
Also, represent them in Venn diagrams.
b) Let P = { x: x ∈ â„• and x < 10 } and Q = { y: y is a factor of 8 }, find
(i) P ∪ Q = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }
(ii) P ∩ Q = { 1, 2, 4, 8 }
(iii) P - Q = { 3, 5, 6, 7, 9 }
(iv) Q - P = { }
Also, show these operations in Venn diagrams.
c) If M = { x: x is an odd number between 10 and 20 } and N = { y: y is a prime number between 15 and 25 }, find
(i) M ∪ N = { 11, 13, 15, 17, 19, 23 }
(ii) M ∩ N = { 17, 19 }
(iii) M - N = { 11, 13, 15 }
(iv) N - M = { 23 }
Show these operations in Venn diagrams.
5. a) Assuming that A and B are two overlapping sets, draw two separate Venn-diagrams to verify A ∪ B = B ∪ (A – B) by shading.
b) Let P and Q are two overlapping sets. Draw two separate Venn-diagrams of P – Q and P – (P ∩ Q) and verify P – Q = P – (P ∩ Q) by shading.
a) If A = {1, 2, 4, 8} and B = {4, 6, 8, 10}, find (A – B) ∪ (B – A).
Solution:Here,
A = {1, 2, 4, 8} and B = {4, 6, 8, 10}
Now, A – B = {1, 2, 4, 8} – {4, 6, 8, 10} = {1, 2}
Again, B – A = {4, 6, 8, 10} – {1, 2, 4, 8} = {6, 10}
∴ (A – B) ∪ (B – A) = {1, 2} ∪ {6, 10} = {1, 2, 6, 10}
b) Find the symmetric difference between the following sets.
(i) A = {m, a, t, h} and B = {m, i, n, d, e, r}
(ii) P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 11}
Solution:
Here,
(i) A = {m, a, t, h} and B = {m, i, n, d, e, r}
Now, A – B = {m, a, t, h} – {m, i, n, d, e, r} = {a, t, h}
Again, B – A = {m, i, n, d, e, r} – {m, a, t, h} = {i, n, d, e, r}
∴ A Δ B = (A – B) ∪ (B – A) = {a, t, h} ∪ {i, n, d, e, r} = {a, d, e, h, i, n, r, t}
(ii) P = {2, 3, 5, 7, 11} and Q = {1, 3, 5, 11}
Now, P – Q = {2, 3, 5, 7, 11} – {1, 3, 5, 11} = {2, 7}
Again, Q – P = {1, 3, 5, 11} – {2, 3, 5, 7, 11} = {1}
∴ P Δ Q = (P – Q) ∪ (Q – P) = {2, 7} ∪ {1} = {1, 2, 7}
7.
a) If U = {1, 2, 3, ..., 10}, A ={1, 3, 5, 7, 9} and B = {2, 3, 5, 7}, find the following sets.(i) A̅
(ii) B̅
(iii) A̅ ∪ B̅
(iv) A̅ ∩ B̅
(v) A ∪ B
(vi) A ∩ B
(vii) A̿
(viii) B̿
Solution:
Here, U = {1, 2, 3, ..., 10}, A = {1, 3, 5, 7, 9} and B = {2, 3, 5, 7}
(i) A̅ = {2, 4, 6, 8, 10}
(ii) B̅ = {1, 4, 6, 8, 9, 10}
(iii) A̅ ∪ B̅ = {1, 2, 4, 6, 8, 9, 10}
(iv) A̅ ∩ B̅ = {4, 6, 8}
(v)A ∪ B = {1, 2, 3, 5, 7, 9}
(vi) A ∩ B = {3, 5, 7}
(vii) A̿ = {1, 3, 5, 7}
(viii) B̿ = {2, 3, 5, 7}
b) If U = {1, 2, 3, ..., 15} and A = {2, 4, 6, 8, 10, 12, 14}, find:
(i) A̅
(ii) A ∪ A̅
(iii) A ∩ A̅
(iv) U̅
Solution:
Here, U = {1, 2, 3, ..., 15} and A = {2, 4, 6, 8, 10, 12, 14}
(i) A̅ = U – A = {1, 3, 5, 7, 9, 11, 13, 15}
(ii) A ∪ A̅ = {2, 4, 6, 8, 10, 12, 14} ∪ {1, 3, 5, 7, 9, 11, 13, 15} = {1, 2, 3, ..., 15} = U
(iii) A ∩ A̅ = ={2, 4, 6, 8, 10, 12, 14} ∩ {1, 3, 5, 7, 9, 11, 13, 15}= ∅
(iv) U̅ = U – U = {1, 2, 3, …, 15} – {1, 2, 3, …, 15} = ∅
Creative section
8.
a) If U = {1, 2, 3, ..., 10}, P = {1, 2, 3, 4, 5} and Q = {2, 4, 6, 8}, list the elements of the following set operations and represent them by shading in Venn-diagrams.
(i) P ∪ Q and P ∪ Q
(ii) P ∩ Q and P ∩ Q
(iii) P – Q and P – Q
(iv) Q – P and Q – P
(v) P̅ ∪ Q̅
(vi) P̅ ∩ Q̅
Solution:
Here, U = {1, 2, 3, ..., 10}, P = {1, 2, 3, 4, 5} and Q = {2, 4, 6, 8}
(i) P ∪ Q = {1, 2, 3, 4, 5} ∪ {2, 4, 6, 8}
= {1, 2, 3, 4, 5, 6, 8}
P ∪ Q = U - (P ∪ Q)
= {1, 2, 3, ... 10} - {1, 2, 3, 4, 5, 6, 8}
= {7, 9, 10}
(ii) P ∩ Q = {1, 2, 3, 4, 5} ∩ {2, 4, 6, 8}
= {2, 4}
P ∩ Q = U - (P ∩ Q)
= {1, 2, 3, ... 10} - {2, 4}
= {1, 3, 5, 6, 7, 8, 9, 10}
(iii) P - Q = {1, 2, 3, 4, 5} - {2, 4, 6, 8}
= {1, 3, 5}
P - Q = U - (P - Q)
= {1, 2, 3, ... 10} - {1, 3, 5}
= {2, 4, 6, 7, 8, 9, 10}
(iv) Q - P = {2, 4, 6, 8} - {1, 2, 3, 4, 5}
= {6, 8}
Q - P = U - (Q - P)
= {1, 2, 3, ... 10} - {6, 8}
= {1, 2, 3, 4, 5, 7, 9, 10}
(v) P̅ = U - P = {6, 7, 8, 9, 10} and
Q̅ = U - Q = {1, 3, 5, 7, 9, 10}
P̅ ∪ Q̅ = {6, 7, 8, 9, 10} ∪ {1, 3, 5, 7, 9, 10}
= {1, 3, 5, 6, 7, 8, 9, 10}
(vi) P̅ ∩ Q̅
= {6, 7, 8, 9, 10} ∩ {1, 3, 5, 7, 9, 10}
= {7, 9, 10}
b) A = {1, 3, 5, 7, 9, 11}, B = {1, 2, 3, 4, 5, 6, 7}, and C = {3, 6, 9, 12, 15} are subsets of the universal set U = {1, 2, 3, ..., 15}. List the elements of the following set operations and illustrate them in Venn diagrams by shading.
(i) A ∪ B ∪ C and A ∪ B ∪ C
(ii) A ∩ B ∩ C and A ∩ B ∩ C
(iii) (A ∪ B) ∩ C and (A ∪ B) ∩ C
(iv) A ∩ (B ∪ C) and A ∩ (B ∪ C)
(v) (A - B) ∪ C and (A - B) ∪ C
(vi) A ∪ (B - C) and A ∪ (B - C)
Solution:
U = {1, 2, 3, ... 15},
A = {1, 3, 5, 7, 9, 11},
B = {1, 2, 3, 4, 5, 6, 7},
C = {3, 6, 9, 12, 15}
(i) A ∪ B ∪ C
= {1, 3, 5, 7, 9, 11} ∪ {1, 2, 3, 4, 5, 6, 7} ∪ {3, 6, 9, 12, 15}
= {1, 2, 3, 4, 5, 6, 7, 9, 11, 12, 15}
A ∪ B ∪ C
= U - (A ∪ B ∪ C)
= {8, 10, 13, 14}
(ii) A ∩ B ∩ C
= {1, 3, 5, 7, 9, 11} ∩ {1, 2, 3, 4, 5, 6, 7} ∩ {3, 6, 9, 12, 15}
= {3}
A ∩ B ∩ C = U - (A ∩ B ∩ C)
= {1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
(iii) (A ∪ B) ∩ C
A ∪ B = {1, 3, 5, 7, 9, 11} ∪ {1, 2, 3, 4, 5, 6, 7}
= {1, 2, 3, 4, 5, 6, 7, 9, 11}
(A ∪ B) ∩ C = {1, 2, 3, 4, 5, 6, 7, 9, 11} ∩ {3, 6, 9, 12, 15}
= {3, 6, 9}
(A ∪ B) ∩ C = U - [(A ∪ B) ∩ C]
= {1, 2, 3, ... 15} - {3, 6, 9}
= {1, 2, 4, 5, 7, 8, 10, 11, 12, 13, 14, 15}
(iv) A ∩ (B ∪ C)
B ∪ C = {1, 2, 3, 4, 5, 6, 7} ∪ {3, 6, 9, 12, 15}
= {1, 2, 3, 4, 5, 6, 7, 9, 12, 15}
A ∩ (B ∪ C) = {1, 3, 5, 7, 9, 11} ∩ {1, 2, 3, 4, 5, 6, 7, 9, 12, 15}
= {1, 3, 5, 7, 9}
(A ∪ B) ∩ C = U - [A ∩ (B ∪ C)]
= {1, 2, 3, ... 15} - {1, 3, 5, 7, 9}
= {2, 4, 6, 8, 10, 11, 12, 13, 14, 15}
(v) (A - B) ∪ C
A - B = {1, 3, 5, 7, 9, 11} - {1, 2, 3, 4, 5, 6, 7}
= {9, 11}
(A - B) ∪ C = {9, 11} ∪ {3, 6, 9, 12, 15}
= {3, 6, 9, 11, 12, 15}
(A - B) ∪ C = U - (A - B) ∪ C
= {1, 2, 3, ... 15} - {3, 6, 9, 11, 12, 15}
= {1, 2, 4, 5, 7, 8, 10, 13, 14}
(vi) A ∪ (B - C)
B - C = {1, 2, 3, 4, 5, 6, 7} - {3, 6, 9, 12, 15}
= {1, 2, 4, 5, 7}
A ∪ (B - C) = {1, 3, 5, 7, 9, 11} ∪ {1, 2, 4, 5, 7}
= {1, 2, 3, 4, 5, 7, 9, 11}
A ∪ (B - C) = {1, 2, 3, ..., 15} - {1, 2, 3, 4, 5, 7, 9, 11}
= {6, 8, 10, 12, 13, 14, 15}
9. a) If U = {0, 1, 2, ..., 10}, A = {2, 3, 5, 7} and B = {1, 3, 5, 7, 9}, verify the following operations.
(i) A ∪ B = A̅ ∩ B̅
(ii) A ∩ B = A̅ ∪ B̅
Solution:
Here, U = {0, 1, 2, ..., 10}, A = {2, 3, 5, 7}, and B = {1, 3, 5, 7, 9}
(i) To verify: A ∪ B = A̅ ∩ B̅
Now, A ∪ B = {2, 3, 5, 7} ∪ {1, 3, 5, 7, 9} = {1, 2, 3, 5, 7, 9}
∴ A ∪ B = U - (A ∪ B) = {0, 1, 2, ..., 10} - {1, 2, 3, 5, 7, 9}
= {0, 4, 6, 8, 10} ... (1)
Again, A̅ = U - A = {0, 1, 2, ..., 10} - {2, 3, 5, 7} = {0, 1, 4, 6, 8, 9, 10}
B̅ = U - B = {0, 1, 2, ..., 10} - {1, 3, 5, 7, 9} = {0, 2, 4, 6, 8, 10}
∴ A̅ ∩ B̅ = {0, 1, 4, 6, 8, 9, 10} ∩ {0, 2, 4, 6, 8, 10}
= {0, 4, 6, 8, 10} ... (2)
From (1) and (2), we get A ∪ B = A̅ ∩ B̅ ✅ verified.
(ii) To verify: A ∩ B = A̅ ∪ B̅
Now, A ∩ B = {2, 3, 5, 7} ∩ {1, 3, 5, 7, 9} = {3, 5, 7}
∴ A ∩ B = U - (A ∩ B) = {0, 1, 2, ..., 10} - {3, 5, 7}
= {0, 1, 2, 4, 6, 8, 9, 10} ... (1)
Again, A̅ = U - A = {0, 1, 2, ..., 10} - {2, 3, 5, 7} = {0, 1, 4, 6, 8, 9, 10}
B̅ = U - B = {0, 1, 2, ..., 10} - {1, 3, 5, 7, 9} = {0, 2, 4, 6, 8, 10}
∴ A̅ ∪ B̅ = {0, 1, 4, 6, 8, 9, 10} ∪ {0, 2, 4, 6, 8, 10}
= {0, 1, 2, 4, 6, 8, 9, 10} ... (2)
From (1) and (2), we get A ∩ B = A̅ ∪ B̅ ✅ verified.
(ii) To verify: A ∆ B = Ā ∆ B̄
Now, A - B = {2, 4, 6, 8} - {3, 6, 9} = {2, 4, 8}
B - A = {3, 6, 9} - {2, 4, 6, 8} = {3, 9}
∴ A ∆ B = (A - B) ∪ (B - A) = {2, 4, 8} ∪ {3, 9} = {2, 3, 4, 8, 9} ... (1)
Again, Ā = U - A = {0, 1, 2, ..., 10} - {2, 3, 5, 7}
= {1, 4, 6, 8, 9, 10}
B̄ = U - B = {0, 1, 2, ..., 10} - {1, 3, 5, 7, 9}
= {2, 4, 6, 8, 10}
∴ Ā ∆ B̄ = (Ā - B̄) ∪ (B̄ - Ā) = {3, 9} ∪ {2, 4, 8}
= {2, 3, 4, 8, 9} ... (2)
From (1) and (2), we get A ∆ B = Ā ∆ B̄ ✅ verified.
10. a) If P = {1, 2, 3, 4, 5, 6}, Q = {2, 4, 6, 8}, and R = {3, 6, 9, 12}, explore the relationship between the following operations.
(i) P ∪ (Q ∩ R) and (P ∪ Q) ∩ (P ∪ R)
(ii) P ∩ (Q ∪ R) and (P ∩ Q) ∪ (P ∩ R)
(iii) P - (Q ∪ R) and (P - Q) ∩ (P - R)
(iv) P - (Q ∩ R) and (P - Q) ∪ (P - R)
Solution:
Here, P = {1, 2, 3, 4, 5, 6}, Q = {2, 4, 6, 8}, and R = {3, 6, 9, 12}
(i)
Q ∩ R = {6}
P ∪ (Q ∩ R) = {1, 2, 3, 4, 5, 6} ... (A)
(P ∪ Q) ∩ (P ∪ R) = {1, 2, 3, 4, 5, 6} ... (B)
From (A) and (B), we observed that P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R).
(ii)
P ∩ (Q ∪ R) = {2, 3, 4, 6} ... (A)
(P ∩ Q) ∪ (P ∩ R) = {2, 3, 4, 6} ... (B)
From (A) and (B), we observed that P ∩ (Q ∪ R) = (P ∩ Q) ∪ (P ∩ R).
(iii)
P - (Q ∪ R) = {1, 5} ... (A)
(P - Q) ∩ (P - R) = {1, 5} ... (B)
From (A) and (B), we observed that P - (Q ∪ R) = (P - Q) ∩ (P - R).
(iv)
P - (Q ∩ R) = {1, 2, 3, 4, 5} ... (A)
(P - Q) ∪ (P - R) = {1, 2, 3, 4, 5} ... (B)
From (A) and (B), we observed that P - (Q ∩ R) = (P - Q) ∪ (P - R).
11. a) If A = {2, 4, 6, 8, 10} and B = {1, 3, 5, 7, 9} are two disjoint sets, verify that n(A ∪ B) = n(A) + n(B).
Solution:
Here, A = {2, 4, 6, 8, 10} ∴ n(A) = 5 and B = {1, 3, 5, 7, 9} ∴ n(B) = 5
A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} ∴ n(A ∪ B) = 10
Again, n(A ∪ B) = 10 = 5 + 5 = n(A) + n(B)
Hence, n(A ∪ B) = n(A) + n(B). ✅ Proved
b) If A = {2, 3, 5, 7} and B = {1, 2, 3, 4, 6, 12} are two overlapping sets, show that n(A ∪ B) = n(A) + n(B) - n(A ∩ B).
Here, A = {2, 3, 5, 7} ∴ n(A) = 4 and B = {1, 2, 3, 4, 6, 12} ∴ n(B) = 6
A ∪ B = {1, 2, 3, 4, 5, 6, 7, 12} ∴ n(A ∪ B) = 8
A ∩ B = {2, 3} ∴ n(A ∩ B) = 2
n(A ∪ B) = 8 = 4 + 6 - 2 = n(A) + n(B) - n(A ∩ B)
Hence, n(A ∪ B) = n(A) + n(B) - n(A ∩ B) ✅ Proved
EXERCISE 1.2
General section 1. (a) A and B are any two disjoint sets. If n(A) = x and n(B) = y, find n(A ∪ B).Solution:
Here, A and B are any two disjoint sets. n (A) = x and n (B) = y
∴ n (A ∪ B) = n (A) + n (B) = x + y
(b) If n(A) = p, n(B) = q, and n(A ∩ B) = r, show this information in a Venn diagram and prove that n (A ∪ B) = n(A) + n(B) – n(A ∩ B).Solution:
Solution:
Here, n(A) = p, n(B) = q, and n (A ∩ B) = r
From Venn-diagram,
n (A ∪ B) = (p –r) + r + (q – r)
= p + q – r
= n (A) + n (B) – n (A ∩ B)
Hence, n (A ∪ B) = n(A) + n(B) – n(A ∩ B)
c) If X and Y are two overlapping subsets of a universal set U, find the relation n (U), between n(X ∪ Y) and n (X ∪ Y) .
Solution:
Solution:
Here, n (U) = n (X ∪ Y) + n (X ∪ Y) .
d) If A and B are two overlapping subsets of a universal set U, write the relation between n(A), n(A ∩ B) and no(A).Solution:
Solution: Here, n₀ (A) = n (A) – n(A ∩ B) or n (A) = n₀ (A) + n (A ∩ B)
2. From the adjoining Venn-diagram, find the cardinal numbers of the following sets:
a) n (U)
b) n (A)
c) n (B)
d) n (A ∪ B)
e) n (A ∩ B)
f) n (A ∪ B)
g) n (A̅)
h) n (B̅)
i) n₀ (A)
j) n₀ (B)
Solution:
From the Venn diagram:
a) n (U)= 9
b) n (A)= 3
c) n (B)= 4
d) n (A ∪ B)= 5
e) n (A ∩ B) = 2
f) n(A ∪ B) = 4
g) n (A̅) = 6
h) n (B̅) = 5
i) n₀ (A)= 1
j) n₀ (B)= 2
3. a) If n(U) = 65, n(A) = 28, n(B) = 45, and n(A ∩ B) = 20, find
(i) n(A ∪ B)
(ii) n (A ∪ B)
(iii) n₀(A)
(iv) n₀(B)
Solution:
Here, n(U) = 65, n(A) = 28, n(B) = 45, and n(A ∩ B) = 20
(i) n(A ∪ B) = n(A) + n(B) - n(A ∩ B) = 28 + 45 - 20 = 53
(ii) n (A ∪ B) = n(U) - n(A ∪ B) = 65 - 53 = 12
(iii) n₀(A) = n(A) - n(A ∩ B) = 28 - 20 = 8
(iv) n₀(B) = n(B) - n(A ∩ B) = 45 - 20 = 25
b) P and Q are the subsets of a universal set U. If n (P) = 55 %, n (Q) = 50 %, and n (P ∪ Q) = 15 %, find:
(i) n (P ∪ Q)
(ii) n (P ∩ Q)
(iii) n (only P)
(iv) n (only Q)
Solution:
Here, n (U) = 100 %, n (P) = 55 %, n (Q) = 50 %, and n (P̅ ∪ Q̅) = 15 %
(i) n (P ∪ Q) = n (U) - n (P ∪ Q) = 100 - 15 = 85 %
(ii) n (P ∩ Q) = n (P) + n (Q) - n (P ∪ Q) = 55 + 50 - 85 = 20 %
(iii) n (only P) = n (P) - n (P ∩ Q) = 55 - 20 = 35 %
(iv) n(only Q) = n (Q) - n (P ∩ Q) = 50 - 20 = 30 %
c) X and Y are the subsets of a universal set U. If n (U) = 88, n₀ (X) = 35, n₀ (Y) = 30, and n (X ∩ Y) = 10, find:
(i) n (X)
(ii) n (Y)
(iii) n (X ∪ Y)
(iv) n (X ∪ Y))
Solution:
Here, n (U) = 88, n₀ (X) = 35, n₀ (Y) = 30, and n (X ∩ Y) = 10
(i) n (X) = n₀ (X) + n (X ∩ Y) = 35 + 10 = 45
(ii) n (Y) = n₀ (Y) + n (X ∩ Y) = 30 + 10 = 40
(iii) n (X ∪ Y) = n₀ (X) + n₀ (Y) + n (X ∩ Y) = 35 + 30 + 10 = 75
(iv) n (X ∪ Y)) = n (U) - n (X ∪ Y) = 88 - 75 = 13
Creative section4. a) In a survey of 600 people in a village of Dhading district, 400 people said they can speak Tamang language, 350 said Nepali language and 200 of them said they can speak both the languages.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) How many people can speak Tamang language only?
(iii) How many people can speak Nepali language only?
(iv) How many people cannot speak any of two languages?
Solution:
Let T and N denote the sets of people who can speak Tamang and Nepali language respectively.
Then, n (U) = 600, n (T) = 400, n (N) = 350 and n (T ∩ N) = 200
Now,
(i) Representing the above information in a Venn-diagram.
(ii) n₀ (T) = n (T) - n (T ∩ N) = 400 - 200 = 200
∴ 200 people can speak Tamang language only.
(iii) n₀ (N) = n (N) - n (T ∩ N) = 350 - 200 = 150
∴ 150 people can speak Nepali language only.
(iv) From Venn-diagram, n (T ∪ N) = 50
Hence, 50 cannot speak any of two languages.
b) In a survey of 1500 people, 775 of them like Nepal Idol, 975 liked Comedy Champion, and 450 people liked both the shows.
(i) Show the above information in a Venn-diagram.
(ii) How many people did not like both the shows?
Solution:
Let N and C denote the sets of people who like Nepal Idol and Comedy Champion respectively.
Then, n (U) = 1500, n (N) = 775, n (C) = 975 and n (N ∩ C) = 450
Now,
(i) Showing the above information in a Venn-diagram.
(ii) From Venn-diagram, n (N ∪ C) = 200
Hence, 200 people did not like both the shows.
c) In a group of 250 music lovers, 135 of them like folk songs, and 150 like modern songs. By drawing a Venn-diagram, find:
(i) how many people like both the songs?
(ii) How many people like only folk songs?
Solution:
Let F and M denote the sets of people who like folk and modern songs respectively.
Then, n (U) = 250, n (F) = 135, n (M) = 150.
Let, n (F ∩ M) = x
Now,
(i) From a Venn-diagram, (135 - x) + x + (150 - x) = 250
or, 285 - x = 250
or, x = 35
Thus, 35 people like both the songs.
(ii) n₀ (F) = 135 - x = 135 - 35
5. a) In a group of 500 students, 280 like bananas, 310 like apples, and 55 do not like both the fruits.
(i) Find the number of students who like both the fruits.
(ii) Find the number of students who like only one fruit.
(iii) Show the result in a Venn-diagram.
Solution:
Let A and B denote the sets of students who like apples and bananas respectively.
Then, n (U) = 500, n (A) = 310, n (B) = 280 and n (A ∪ B) = 55
(i) We have, n (A ∪ B) = n (U) - n (A ∪ B) = 500 - 55 = 445
Also, n (A ∩ B) = n (A) + n (B) - n (A ∪ B)
= 310 + 280 - 445
= 145
So, 145 students like both the fruits.
(ii) The number of students who like only one fruit = n₀ (A) + n₀ (B)
= n (A ∪ B) - n (A ∩ B)
= 445 - 145
= 300
(iii) Showing the above information in a Venn-diagram.
b) In a survey of 900 students in a school, it was found that 600 students liked tea, 500 liked coffee, and 125 did not like both drinks.
(i) Draw a Venn-diagram to illustrate the above information.
(ii) Find the number of students who like both drinks.
(iii) Find the number of students who like one of these drinks only.
Solution:
Let T and C denote the sets of students who like tea and coffee respectively.
Then, n (U) = 900, n (T) = 600, n (C) = 500 and n (T ∪ C) = 125
Let, n (T ∩ C) = x
Now,
(i) Drawing the above information in a Venn-diagram.
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