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Q3:
$A = \{1, 2, 3, 5\}$ and $B = \{4, 6, 9\}$. Define a relation $R$ from $A$ to $B$ by $R = \{(x, y): \text{the difference between } x \text{ and } y \text{ is odd}; x \in A, y \in B\}$. Write $R$ in roster form.
Solution :
Step 1: Establishing the Domain and Codomain
Given the sets:
- $A = \{1, 2, 3, 5\}$
- $B = \{4, 6, 9\}$
The relation $R$ is defined as $R = \{(x, y) : |x - y| \in \{ \text{odd numbers} \}; x \in A, y \in B\}$. [Per the definition of relation in Cartesian product $A \times B$].
Step 2: Systematic Evaluation of Ordered Pairs $(x, y)$
To determine if the difference $|x - y|$ is odd, we utilize the parity property: The difference between two integers is odd if and only if one integer is even and the other is odd. [Parity Law: $Odd - Even = Odd$ and $Even - Odd = Odd$].
| $x \in A$ | $y \in B$ | Difference $|x - y|$ | Parity | Include in $R$? |
|---|---|---|---|---|
| 1 | 4 | $|1-4| = 3$ | Odd | Yes |
| 1 | 6 | $|1-6| = 5$ | Odd | Yes |
| 1 | 9 | $|1-9| = 8$ | Even | No |
| 2 | 4 | $|2-4| = 2$ | Even | No |
| 2 | 6 | $|2-6| = 4$ | Even | No |
| 2 | 9 | $|2-9| = 7$ | Odd | Yes |
| 3 | 4 | $|3-4| = 1$ | Odd | Yes |
| 3 | 6 | $|3-6| = 3$ | Odd | Yes |
| 3 | 9 | $|3-9| = 6$ | Even | No |
| 5 | 4 | $|5-4| = 1$ | Odd | Yes |
| 5 | 6 | $|5-6| = 1$ | Odd | Yes |
| 5 | 9 | $|5-9| = 4$ | Even | No |
Step 3: Visual Representation of the Relation
Step 4: Writing the Roster Form
Collecting all identified pairs $(x, y)$ where the difference is odd:
$R = \{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)\}$
Final Solution: $R = \{(1, 4), (1, 6), (2, 9), (3, 4), (3, 6), (5, 4), (5, 6)\}$
More Questions from Class 11 Mathematics Relations and Functions EXERCISE 2.2
- Q1: Let $A = \{1, 2, 3,...,14\}$. Define a relation $R$ from $A$ to $A$ by $R = \{(x, y) : 3x – y = 0, \text{where } x, y \in A\}$. Write down its domain, codomain and range.
- Q2: Define a relation $R$ on the set $\mathbb{N}$ of natural numbers by $R = \{(x, y) : y = x + 5, x \text{ is a natural number less than } 4; x, y \in \mathbb{N}\}$. Depict this relationship using roster form. Write down the domain and the range.
- Q4(i): The Fig2.7 shows a relationship between the sets $P$ and $Q$. Write this relation (i) in set-builder form. What is its domain and range?
- Q4(ii): The Fig2.7 shows a relationship between the sets $P$ and $Q$. Write this relation (ii) roster form. What is its domain and range?
- Q5(i): Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b): a, b \in A, b \text{ is exactly divisible by } a\}$. (i) Write $R$ in roster form.
- Q5(ii): Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b): a, b \in A, b \text{ is exactly divisible by } a\}$. (ii) Find the domain of $R$.
- Q5(iii): Let $A = \{1, 2, 3, 4, 6\}$. Let $R$ be the relation on $A$ defined by $\{(a, b): a, b \in A, b \text{ is exactly divisible by } a\}$. (iii) Find the range of $R$.
- Q6: Determine the domain and range of the relation $R$ defined by $R = \{(x, x + 5) : x \in \{0, 1, 2, 3, 4, 5\}\}$.
- Q7: Write the relation $R = \{(x, x^3) : x \text{ is a prime number less than } 10\}$ in roster form.
- Q8: Let $A = \{x, y, z\}$ and $B = \{1, 2\}$. Find the number of relations from $A$ to $B$.
- Q9: Let $R$ be the relation on $\mathbb{Z}$ defined by $R = \{(a,b): a, b \in \mathbb{Z}, a – b \text{ is an integer}\}$. Find the domain and range of $R$.
CBSE Solutions for Class 11 Mathematics Relations and Functions
Chapters in CBSE - Class 11 Mathematics
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