10 maths chapter 1 || 1.2 || Samacheer Kalvi maths

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 \( B \)?

Concept: A relation from set \( A \) to set \( B \) must be a subset of the Cartesian product \( A \times B \). That is, every ordered pair \((x, y)\) in the relation must have \( x \in A \) and \( y \in B \).

Given sets: \( A = \{1, 2, 3, 7\} \) and \( B = \{3, 0, -1, 7\} \)

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

Here, the element \( 1 \notin B \). Therefore, \( (2, 1) \notin A \times B \).

Result: Not a relation from \( A \) to \( B \).

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

Here, the element \( -1 \notin A \) and \( 1 \notin B \). Therefore, \( (-1, 1) \notin A \times B \).

Result: Not a relation from \( A \) to \( B \).

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

Check each ordered pair: \( 2 \in A, -1 \in B \); \( 7 \in A, 7 \in B \); \( 1 \in A, 3 \in B \). All pairs belong to \( A \times B \).

Result: It is a relation from \( A \) to \( B \).

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

Here, the element \( 0 \notin A \). Therefore, \( (0, 3) \notin A \times B \) and \( (0, 7) \notin A \times B \).

Result: Not a relation from \( A \) to \( B \).

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

Given \( A = \{1, 2, 3, 4, \dots, 45\} \).

The relation \( R \) is "square of a number", meaning \( y = x^2 \) where \( x, y \in A \).

Let's find the valid pairs where \( x^2 \le 45 \):

• If \( x = 1 \), \( y = 1^2 = 1 \implies (1, 1) \)
• If \( x = 2 \), \( y = 2^2 = 4 \implies (2, 4) \)
• If \( x = 3 \), \( y = 3^2 = 9 \implies (3, 9) \)
• If \( x = 4 \), \( y = 4^2 = 16 \implies (4, 16) \)
• If \( x = 5 \), \( y = 5^2 = 25 \implies (5, 25) \)
• If \( x = 6 \), \( y = 6^2 = 36 \implies (6, 36) \)
• If \( x = 7 \), \( y = 7^2 = 49 \notin A \) (Stop here since 49 exceeds 45)

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

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

Range of \( R \): Set of second coordinates = \(\{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.

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

Let's compute \( y \) for each value of \( x \):

• For \( x = 0 \), \( y = 0 + 3 = 3 \)
• For \( x = 1 \), \( y = 1 + 3 = 4 \)
• For \( x = 2 \), \( y = 2 + 3 = 5 \)
• For \( x = 3 \), \( y = 3 + 3 = 6 \)
• For \( x = 4 \), \( y = 4 + 3 = 7 \)
• For \( x = 5 \), \( y = 5 + 3 = 8 \)

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

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 conditions: \( x = 2y \), where \( x \in \{2, 3, 4, 5\} \) and \( y \in \{1, 2, 3, 4\} \).

Test values of \( y \):

• If \( y = 1 \), \( x = 2(1) = 2 \) (Since \( 2 \in \{2, 3, 4, 5\} \), pair is \((2, 1)\))
• If \( y = 2 \), \( x = 2(2) = 4 \) (Since \( 4 \in \{2, 3, 4, 5\} \), pair is \((4, 2)\))
• If \( y = 3 \), \( x = 2(3) = 6 \) (Since \( 6 \notin \{2, 3, 4, 5\} \), not valid)
• If \( y = 4 \), \( x = 2(4) = 8 \) (Since \( 8 \notin \{2, 3, 4, 5\} \), not valid)

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

(a) Arrow Diagram: Draw two ovals for sets \( X = \{2, 3, 4, 5\} \) and \( Y = \{1, 2, 3, 4\} \). Draw directed arrows from 2 to 1, and from 4 to 2.

(b) Graph: Plot the discrete points \((2, 1)\) and \((4, 2)\) on a Cartesian coordinate plane.

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

Natural numbers less than 10 are \( \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \).

Find pairs where \( y = x + 3 \):

• \( x = 1 \implies y = 4 \)   \((1, 4)\)
• \( x = 2 \implies y = 5 \)   \((2, 5)\)
• \( x = 3 \implies y = 6 \)   \((3, 6)\)
• \( x = 4 \implies y = 7 \)   \((4, 7)\)
• \( x = 5 \implies y = 8 \)   \((5, 8)\)
• \( x = 6 \implies y = 9 \)   \((6, 9)\)
• For \( x \ge 7 \), \( y \ge 10 \), which are not less than 10.

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

(a) Arrow Diagram: Represent sets of numbers from 1 to 9 in two ovals and draw arrows linking \( 1 \to 4, 2 \to 5, 3 \to 6, 4 \to 7, 5 \to 8, 6 \to 9 \).

(b) Graph: Plot the 6 ordered pairs on a Cartesian plane.

5. A company has four categories of employees given by Assistants (\( A \)), Clerks (\( C \)), Managers (\( M \)) and an Executive Officer (\( E \)). Salaries are ₹10,000, ₹25,000, ₹50,000 and ₹1,00,000 respectively. If \( A_1-A_5 \) are Assistants, \( C_1-C_4 \) are Clerks, \( M_1-M_3 \) are managers, and \( E_1-E_2 \) are Executive officers, and relation \( R \) is defined by \( xRy \) where \( x \) is the salary given to person \( y \), express \( R \) through an ordered pair and an arrow diagram.

Given mapping of salaries to categories/persons:

• ₹10,000 is given to Assistants: \( A_1, A_2, A_3, A_4, A_5 \)
• ₹25,000 is given to Clerks: \( C_1, C_2, C_3, C_4 \)
• ₹50,000 is given to Managers: \( M_1, M_2, M_3 \)
• ₹1,00,000 is given to Executive Officers: \( E_1, E_2 \)

The relation \( xRy \) means the first coordinate \( x \) is the salary and the second coordinate \( y \) is the person.

Ordered Pairs (\( R \)):

\( R = \{ (10000, A_1), (10000, A_2), (10000, A_3), (10000, A_4), (10000, A_5),
(25000, C_1), (25000, C_2), (25000, C_3), (25000, C_4),
(50000, M_1), (50000, M_2), (50000, M_3),
(100000, E_1), (100000, E_2) \} \)

Arrow Diagram Description: Two sets (Domain containing salary values: \( \{10000, 25000, 50000, 100000\} \) and Codomain containing all employees: \( \{A_1 \dots E_2\} \)). Draw arrows from salary 10000 to each of \( A_1 \) through \( A_5 \), from 25000 to \( C_1 \) through \( C_4 \), from 50000 to \( M_1 \) through \( M_3 \), and from 100000 to \( E_1 \) and \( E_2 \).