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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?
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?
Solution :
Given Variables & Initial Mapping
Based on the provided relationship diagram (Fig 2.7), we identify the sets $P$ and $Q$ and their corresponding mapping pairs $(x, y)$ where $x \in P$ and $y \in Q$:
- Set $P = \{5, 6, 7\}$
- Set $Q = \{3, 4, 5\}$
- Mapping pairs: $(5, 3), (6, 4), (7, 5)$
Step 1: Identifying the Logical Relation
Observe the numerical correspondence between each input $x \in P$ and output $y \in Q$:
- For $x=5$, $y=3$ $\implies$ $5 - 2 = 3$
- For $x=6$, $y=4$ $\implies$ $6 - 2 = 4$
- For $x=7$, $y=5$ $\implies$ $7 - 2 = 5$
The universal rule governing this relation is $y = x - 2$, or equivalently, $x - y = 2$. [Per the Definition of a Relation as a subset of the Cartesian Product $P \times Q$].
Step 2: Defining Set-Builder Form
The set-builder form defines the relation $R$ by specifying the properties that the elements $(x, y)$ must satisfy.
$R = \{ (x, y) : y = x - 2, x \in P, y \in Q \}$
Step 3: Determining Domain and Range
The Domain is defined as the set of all first components of the ordered pairs in the relation, and the Range is the set of all second components [Per the Set-Theoretic definition of functions/relations].
| Component | Definition | Result |
|---|---|---|
| Domain | $\{x : (x, y) \in R\}$ | $\{5, 6, 7\}$ |
| Range | $\{y : (x, y) \in R\}$ | $\{3, 4, 5\}$ |
Final Solution:
Set-Builder Form: $R = \{ (x, y) : y = x - 2, x \in \{5, 6, 7\}, y \in \{3, 4, 5\} \}$
Domain: $\{5, 6, 7\}$
Range: $\{3, 4, 5\}$
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(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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