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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.
Solution :
Step 1: Establishing the Set of Inputs ($x$)
The relation $R$ is defined on the set of natural numbers $\mathbb{N} = \{1, 2, 3, \dots\}$. The condition states that $x$ must be a natural number less than $4$. Therefore, the possible values for $x$ are:
$x \in \{1, 2, 3\}$ [Definition of Natural Numbers and inequality constraint].
Step 2: Calculating Corresponding Outputs ($y$)
Using the governing equation $y = x + 5$, we determine the image of each $x$ under the relation $R$:
- For $x = 1$: $y = 1 + 5 = 6$
- For $x = 2$: $y = 2 + 5 = 7$
- For $x = 3$: $y = 3 + 5 = 8$
Since $\{6, 7, 8\} \subset \mathbb{N}$, all calculated pairs satisfy the condition $x, y \in \mathbb{N}$ [Closure property of natural numbers under addition].
Step 3: Representing the Relation in Roster Form
The roster form of a relation is the set of all ordered pairs $(x, y)$ that satisfy the defined rule.
$R = \{(1, 6), (2, 7), (3, 8)\}$
Step 4: Determining Domain and Range
The Domain is the set of all first elements of the ordered pairs in $R$, and the Range is the set of all second elements of the ordered pairs in $R$.
- Domain: $\{1, 2, 3\}$
- Range: $\{6, 7, 8\}$
Step 5: Visual Representation of the Relation
Final Solution: The relation in roster form is $R = \{(1, 6), (2, 7), (3, 8)\}$. The Domain is $\{1, 2, 3\}$ and the Range is $\{6, 7, 8\}$.
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.
- 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 $B$.
- 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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