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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 $R$.
Solution :
Step 1: Analyzing the Definition of the Relation
The relation $R$ is defined on the set of integers $\mathbb{Z}$ as follows:
$R = \{(a, b) : a, b \in \mathbb{Z}, a - b \in \mathbb{Z}\}$
[By the property of closure under subtraction for integers: If $a, b \in \mathbb{Z}$, then $(a - b)$ is always an integer $\mathbb{Z}$.]
Step 2: Determining the Domain of R
The domain of a relation $R$ consists of all the first elements of the ordered pairs $(a, b)$ that belong to $R$.
- For any integer $a \in \mathbb{Z}$, we must determine if there exists an integer $b \in \mathbb{Z}$ such that $(a, b) \in R$.
- Per the definition, $(a, b) \in R$ if and only if $a - b = k$, where $k \in \mathbb{Z}$.
- Let $b = a - k$. Since $a \in \mathbb{Z}$ and $k \in \mathbb{Z}$, $b$ is necessarily an integer [$b \in \mathbb{Z}$].
- Therefore, every integer $a$ can be associated with at least one integer $b$ (in fact, infinitely many) such that $(a, b) \in R$.
Thus, the domain of $R$ is the set of all integers, $\mathbb{Z}$.
Step 3: Determining the Range of R
The range of a relation $R$ consists of all the second elements of the ordered pairs $(a, b)$ that belong to $R$.
- For any integer $b \in \mathbb{Z}$, we must determine if there exists an integer $a \in \mathbb{Z}$ such that $(a, b) \in R$.
- Per the definition, $(a, b) \in R$ if and only if $a - b = k$, where $k \in \mathbb{Z}$.
- Let $a = b + k$. Since $b \in \mathbb{Z}$ and $k \in \mathbb{Z}$, $a$ is necessarily an integer [$a \in \mathbb{Z}$].
- Therefore, every integer $b$ is the image of at least one integer $a$ under the relation $R$.
Thus, the range of $R$ is the set of all integers, $\mathbb{Z}$.
Step 4: Logical Mapping Illustration
| Set | Definition |
|---|---|
| Domain | $\{x : x \in \mathbb{Z}\}$ |
| Range | $\{y : y \in \mathbb{Z}\}$ |
Final Solution: The domain of $R$ is $\mathbb{Z}$ and the range of $R$ is $\mathbb{Z}$.
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?
- 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 $B$.
CBSE Solutions for Class 11 Mathematics Relations and Functions
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