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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?
The Fig2.7 shows a relationship between the sets $P$ and $Q$. Write this relation (ii) roster form. What is its domain and range?
Solution :
Given Data & Visual Interpretation
Based on the standard representation of the provided function diagram (Figure 2.7), we observe the following mapping between set $P$ and set $Q$:
- Set $P = \{5, 6, 7\}$
- Set $Q = \{3, 4, 5\}$
- The arrows indicate mapping as follows: $5 \to 3$, $6 \to 4$, $7 \to 5$.
Step 1: Expressing the Relation in Roster Form
A relation in roster form is represented as a set of ordered pairs $(x, y)$ such that $x \in P$ and $y \in Q$, and $x$ is related to $y$ [Per Definition of a Relation in Set Theory].
From the visual mapping:
- $5$ is related to $3 \implies (5, 3)$
- $6$ is related to $4 \implies (6, 4)$
- $7$ is related to $5 \implies (7, 5)$
Thus, the Roster Form $R$ is:
$R = \{(5, 3), (6, 4), (7, 5)\}$
Step 2: Determining the Domain
The domain of a relation is the set of all first components of the ordered pairs in $R$ [Per Definition of Domain].
Extracting the first components from $R$:
$\text{Domain} = \{5, 6, 7\}$
Step 3: Determining the Range
The range of a relation is the set of all second components of the ordered pairs in $R$ [Per Definition of Range].
Extracting the second components from $R$:
$\text{Range} = \{3, 4, 5\}$
Final Solution: The relation in roster form is $\{(5, 3), (6, 4), (7, 5)\}$. The domain is $\{5, 6, 7\}$ and the range is $\{3, 4, 5\}$.
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.
- 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.
- 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?
- 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
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