10 maths 1.2

Exercise 1.2 Solutions

1. Let \(A = \{1,2,3,7\}\) and \(B = \{3,0,-1,7\}\), which of the following are relations from \(A\) to \(B\)?

Note: A relation \(R\) from \(A\) to \(B\) must be a subset of \(A \times B\) (\(R \subseteq A \times B\)).

(i) \(R_1 = \{(2,1), (7,1)\}\)

Here, \(1 \notin B\). So, \((2,1) \notin A \times B\) and \((7,1) \notin A \times B\).

\(R_1\) is NOT a relation from \(A\) to \(B\).

(ii) \(R_2 = \{(-1,1)\}\)

Here, \(-1 \notin A\). So, \((-1,1) \notin A \times B\).

\(R_2\) is NOT a relation from \(A\) to \(B\).

(iii) \(R_3 = \{(2,-1), (7,7), (1,3)\}\)

Here, \((2,-1) \in A \times B\), \((7,7) \in A \times B\), and \((1,3) \in A \times B\).

\(R_3 \subseteq A \times B\). So, \(R_3\) IS a relation from \(A\) to \(B\).

(iv) \(R_4 = \{(7,-1), (0,3), (3,3), (0,7)\}\)

Here, \(0 \notin A\). So, \((0,3) \notin A \times B\) and \((0,7) \notin A \times B\).

\(R_4\) is NOT a relation from \(A\) to \(B\).

2. Let \(A = \{1,2,3,4,\dots,45\}\) and \(R\) be the relation defined as "square is of a number" on \(A\). Write \(R\) as a subset of \(A \times A\). Also, find the domain and range of \(R\).

Solution:

Given \(A = \{1,2,3,\dots,45\}\).

The relation \(R = \{(x,y) \mid x \in A, y \in A \text{ and } y = x^2\}\).

Squares of numbers in \(A\): \(1^2=1, 2^2=4, 3^2=9, 4^2=16, 5^2=25, 6^2=36\). (Note: \(7^2=49 \notin A\)).

Relation \(R\): \(R = \{(1,1), (2,4), (3,9), (4,16), (5,25), (6,36)\}\)

Domain of \(R\): \(\{1, 2, 3, 4, 5, 6\}\)

Range of \(R\): \(\{1, 4, 9, 16, 25, 36\}\)

3. A Relation \(R\) is given by the set \(\{(x,y) \mid y = x + 3, x \in \{0,1,2,3,4,5\}\}\). Determine its domain and range.

Solution:

Given \(x \in \{0,1,2,3,4,5\}\) and \(y = x + 3\):

When \(x = 0 \implies y = 0 + 3 = 3\)

When \(x = 1 \implies y = 1 + 3 = 4\)

When \(x = 2 \implies y = 2 + 3 = 5\)

When \(x = 3 \implies y = 3 + 3 = 6\)

When \(x = 4 \implies y = 4 + 3 = 7\)

When \(x = 5 \implies y = 5 + 3 = 8\)

\(R = \{(0,3), (1,4), (2,5), (3,6), (4,7), (5,8)\}\)

Domain: \(\{0, 1, 2, 3, 4, 5\}\)

Range: \(\{3, 4, 5, 6, 7, 8\}\)

4. Represent each of the given relations by (a) an arrow diagram, (b) a graph and (c) a set in roster form, wherever possible.

(i) \(\{(x,y) \mid x = 2y, x \in \{2,3,4,5\}, y \in \{1,2,3,4\}\}\)

Given \(x = 2y \implies y = \frac{x}{2}\):

If \(x=2 \implies y=1 \in \{1,2,3,4\}\)

If \(x=4 \implies y=2 \in \{1,2,3,4\}\)

(For \(x=3, 5\), corresponding \(y\) values are not in the given set).

(c) Set in Roster form: \(R = \{(2,1), (4,2)\}\)

(a) Arrow Diagram: Map \(2 \to 1\) and \(4 \to 2\).

(b) Graph: Plot the points \((2,1)\) and \((4,2)\) on an X-Y plane.

(ii) \(\{(x,y) \mid y = x + 3, x, y \text{ are natural numbers } < 10\}\)

Here \(x, y \in \{1,2,3,4,5,6,7,8,9\}\):

\(x=1 \implies y=4\),   \(x=2 \implies y=5\),   \(x=3 \implies y=6\)

\(x=4 \implies y=7\),   \(x=5 \implies y=8\),   \(x=6 \implies y=9\)

(c) Set in Roster form: \(R = \{(1,4), (2,5), (3,6), (4,7), (5,8), (6,9)\}\)

(a) Arrow Diagram: Map \(1\to 4, 2\to 5, 3\to 6, 4\to 7, 5\to 8, 6\to 9\).

(b) Graph: Plot the points \((1,4), (2,5), (3,6), (4,7), (5,8), (6,9)\).

5. A company has four categories of employees given by Assistants (\(A\)), Clerks (\(C\)), Managers (\(M\)) and an Executive Officer (\(E\)). The company provides ₹10,000, ₹25,000, ₹50,000 and ₹1,000,000 as salaries to the people who work in the categories \(A, C, M\) and \(E\) respectively. If \(A_1, A_2, A_3, A_4, A_5\) were Assistants; \(C_1, C_2, C_3, C_4\) were Clerks; \(M_1, M_2, M_3\) were Managers and \(E_1, E_2\) were Executive officers and if the relation \(R\) is defined by \(xRy\), where \(x\) is the salary given to person \(y\), express the relation \(R\) through an ordered pair and an arrow diagram.

Solution:

Salaries: ₹10,000 (\(A\)), ₹25,000 (\(C\)), ₹50,000 (\(M\)), ₹1,00,000 (\(E\)).

(i) Ordered Pairs:

\(R = \{ (10000, A_1), (10000, A_2), (10000, A_3), (10000, A_4), (10000, A_5),\)

        \((25000, C_1), (25000, C_2), (25000, C_3), (25000, C_4),\)

        \((50000, M_1), (50000, M_2), (50000, M_3),\)

        \((100000, E_1), (100000, E_2) \}\)

(ii) Arrow Diagram:

Set X (Salaries): \(\{10000, 25000, 50000, 100000\}\)

Set Y (Employees): \(\{A_1, A_2, A_3, A_4, A_5, C_1, C_2, C_3, C_4, M_1, M_2, M_3, E_1, E_2\}\)

Arrows point from each salary to its corresponding set of employees.