Balbharati Maharashtra State Board 11th Commerce Maths Solution Book Pdf Chapter 1 Sets and Relations Ex 1.2 Questions and Answers.

## Maharashtra State Board 11th Commerce Maths Solutions Chapter 1 Sets and Relations Ex 1.2

Question 1.

If (x – 1, y + 4) = (1, 2), find the values of x and y.

Solution:

(x – 1, y + 4) = (1, 2)

By the definition of equality of ordered pairs, we have

x – 1 = 1 and y + 4 = 2

∴ x = 2 and y = -2

Question 2.

If \(\left(x+\frac{1}{3}, \frac{y}{3}-1\right)=\left(\frac{1}{3}, \frac{3}{2}\right)\), find x and y.

Solution:

\(\left(x+\frac{1}{3}, \frac{y}{3}-1\right)=\left(\frac{1}{3}, \frac{3}{2}\right)\)

By the definition of equality of ordered pairs, we have

\(x+\frac{1}{3}=\frac{1}{3}\) and \(\frac{y}{3}-1=\frac{3}{2}\)

\(x=\frac{1}{3}-\frac{1}{3}\) and \(\frac{y}{3}=\frac{3}{2}+1=\frac{5}{2}\)

x = 0 and y = \(\frac{15}{2}\)

Question 3.

If A = {a, b, c}, B = {x, y}, find A × B, B × A, A × A, B × B.

Solution:

A = {a, b, c}, B = {x, y}

A × B = {(a, x), (a, y), (b, x), (b, y), (c, x), (c, y)}

B × A = {(x, a), (x, b), (x, c), (y, a), (y, b), (y, c)}

A × A = {(a, a), (a, b), (a, c), (b, a), (b, b), (b, c), (c, a), (c, b), (c, c)}

B × B = {(x, x), (x, y), (y, x), (y, y)}

Question 4.

If P = {1, 2, 3} and Q = {6, 4}, find the sets P × Q and Q × P.

Solution:

P = {1, 2, 3}, Q = {6, 4}

P × Q = {(1, 6), (1, 4), (2, 6), (2, 4), (3, 6), (3, 4)}

Q × P = {(6, 1), (6, 2), (6, 3), (4, 1), (4, 2), (4, 3)}

Question 5.

Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find

(i) A × (B ∩ C)

(ii) (A × B) ∩ (A × C)

(iii) A × (B ∪ C)

(iv) (A × B) ∪ (A × C)

Solution:

A= {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}

(i) B ∩ C = {5, 6}

∴ A × (B ∩ C) = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}

(ii) A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}

A × C = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}

∴ (A × B) ∩ (A × C) = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}

(iii) B ∪ C = {4, 5, 6}

∴ A × (B ∪ C) = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}

(iv) A × B = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}

A × C = {(1, 5), (1, 6), (2, 5), (2, 6), (3, 5), (3, 6), (4, 5), (4, 6)}

∴ (A × B) ∪ (A × C) = {(1, 4), (1, 5), (1, 6), (2, 4), (2, 5), (2, 6), (3, 4), (3, 5), (3, 6), (4, 4), (4, 5), (4, 6)}

Question 6.

Express {(x, y) / x^{2} + y^{2} = 100, where x, y ∈ W} as a set of ordered pairs.

Solution:

{(x, y) / x^{2} + y^{2} = 100, where x, y ∈ W}

We have, x^{2} + y^{2} = 100

When x = 0 and y = 10,

x^{2} + y^{2} = 0^{2} + 10^{2} = 100

When x = 6 andy = 8,

x^{2} + y^{2} = 6^{2} + 8^{2} = 100

When x = 8 and y = 6,

x^{2} + y^{2} = 8^{2} + 6^{2} = 100

When x = 10 and y = 0,

x^{2} + y^{2} = 10^{2} + 0^{2} = 100

∴ Set of ordered pairs = {(0, 10), (6, 8), (8, 6), (10, 0)}

Question 7.

Write the domain and range of the following relations.

(i) {(a, b) / a ∈ N, a < 6 and b = 4}

(ii) {(a, b) / a, b ∈ N, a + b = 12}

(iii) {(2, 4), (2, 5), (2, 6), (2, 7)}

Solution:

(i) Let R_{1} = {(a, b)/ a ∈ N, a < 6 and b = 4}

Set of values of ‘a’ are domain and set of values of ‘b’ are range.

a ∈ N and a < 6

∴ a = 1, 2, 3, 4, 5 and b = 4

Domain (R_{1}) = {1, 2, 3, 4, 5}

Range (R_{1}) = {4}

(ii) Let R_{2} = {(a, b)/a, b ∈ N and a + b = 12}

Now, a, b ∈ N and a + b = 12

When a = 1, b = 11

When a = 2, b = 10

When a = 3, b = 9

When a = 4, b = 8

When a = 5, b = 7

When a = 6, b = 6

When a = 7, b = 5

When a = 8, b = 4

When a = 9, b = 3

When a = 10, b = 2

When a = 11, b = 1

∴ Domain (R_{2}) = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}

Range (R_{2}) = {11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1}

(iii) Let R_{3} = {(2, 4), (2, 5), (2, 6), (2, 7)}

Domain (R_{3}) = {2}

Range (R_{3}) = {4, 5, 6, 7}

Question 8.

Let A = {6, 8} and B = {1, 3, 5}.

Let R = {(a, b) / a ∈ A, b ∈ B, a – b is an even number}.

Show that R is an empty relation from A to B.

Solution:

A= {6, 8}, B = {1, 3, 5}

R = {(a, b)/ a ∈ A, b ∈ B, a – b is an even number}

a ∈ A

∴ a = 6, 8

b ∈ B

∴ b = 1, 3, 5

When a = 6 and b = 1, a – b = 5 which is odd

When a = 6 and b = 3, a – b = 3 which is odd

When a = 6 and b = 5, a – b = 1 which is odd

When a = 8 and b = 1, a – b = 7 which is odd

When a = 8 and b = 3, a – b = 5 which is odd

When a = 8 and b = 5, a – b = 3 which is odd

Thus, no set of values of a and b gives a – b even.

∴ R is an empty relation from A to B.

Question 9.

Write the relation in the Roster form and hence find its domain and range.

(i) R_{1} = {(a, a^{2}) / a is a prime number less than 15}

(ii) R_{2} = {(a, \(\frac{1}{a}\)) / 0 < a ≤ 5, a ∈ N}

Solution:

(i) R_{1} = {(a, a^{2}) / a is a prime number less than 15}

∴ a = 2, 3, 5, 7, 11, 13

∴ a2 = 4, 9, 25, 49, 121, 169

∴ R_{1} = {(2, 4), (3, 9), (5, 25), (7, 49), (11, 121), (13, 169)}

∴ Domain (R_{1}) = {a/a is a prime number less than 15} = {2, 3, 5, 7, 11, 13}

Range (R_{1}) = {a^{2}/a is a prime number less than 15} = {4, 9, 25, 49, 121, 169}

Question 10.

R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}. Find the range of R.

Solution:

R = {(a, b) / b = a + 1, a ∈ Z, 0 < a < 5}

∴ a = 1, 2, 3, 4

∴ b = 2, 3, 4, 5

∴ Range (R) = {2, 3, 4, 5}

Question 11.

Find the following relations as sets of ordered pairs.

(i) {(x, y) / y = 3x, x ∈ {1, 2, 3}, y ∈ {3, 6, 9, 12}}

(ii) {(x,y) / y > x + 1, x ∈ {1, 2} and y ∈ {2, 4, 6}}

(iii) {(x, y) / x + y = 3, x, y ∈ {0, 1, 2, 3}}

Solution:

(i) {(x, y) / y = 3x, x ∈ {1, 2, 3}, y ∈ {3, 6, 9, 12}}

Here y = 3x

When x = 1, y = 3(1) = 3

When x = 2, y = 3(2) = 6

When x = 3, y = 3(3) = 9

∴ Ordered pairs are {(1, 3), (2, 6), (3, 9)}

(ii) {(x, y) / y > x + 1, x ∈ {1, 2} and y ∈ {2, 4, 6}}

Here, y > x + 1

When x = 1 and y = 2, 2 ≯ 1 + 1

When x = 1 and y = 4, 4 > 1 + 1

When x = 1 and y = 6, 6 > 1 + 1

When x = 2 and y = 2, 2 ≯ 2 + 1

When x = 2 and y = 4, 4 > 2 + 1

When x = 2 and y = 6, 6 > 2 + 1

∴ Ordered pairs are {(1, 4), (1, 6), (2, 4), (2, 6)}

(iii) {(x, y) / x + y = 3, x, y ∈ {0, 1, 2, 3}}

Here, x + y = 3

When x = 0, y = 3

When x = 1, y = 2

When x = 2, y = 1

When x = 3, y = 0

∴ Ordered pairs are {(0, 3), (1, 2), (2, 1), (3, 0)}