The relation R defined in `A= {1, 2, 3}` by `aRb` if `|a^2-b^2| leq 5`. Which of the following is faise

Options

A. R = {(1,1),(2,2),(3,3),(2,1),(1,2),(2,3),(3,2)}
B. `R^(-1)=R`
C. Domain of R = {1, 2, 3}
D. Range of R = {5}

Explanation

To solve the problem, we need to analyze the relation \( R \) defined on the set \( A = \{1, 2, 3\} \) by the condition \( aRb \) if \( |a^2 - b^2| \leq 5 \). We will determine the pairs \( (a, b) \) that satisfy this condition and identify which of the given statements is false.

### Step-by-Step Solution:

1. **Understanding the Relation**:
The relation \( R \) is defined such that \( aRb \) holds if the absolute difference of the squares of \( a \) and \( b \) is less than or equal to 5. This can be expressed as:
\[ |a^2 - b^2| \leq 5 \]

2. **Calculating for Each Element in Set \( A \)**:
We will evaluate the relation for each element \( a \) in the set \( A \).

- **For \( a = 1 \)**:
\[ |1^2 - b^2| \leq 5 \implies |1 - b^2| \leq 5 \]
This leads to:
\[ -5 \leq 1 - b^2 \leq 5 \]
Rearranging gives:
\[ -6 \leq -b^2 \leq 4 \implies 0 \leq b^2 \leq 6 \]
The values of \( b \) in \( A \) that satisfy this are \( b = 1, 2 \) (since \( 3^2 = 9 \) which is not ≤ 6).

- **For \( a = 2 \)**:
\[ |2^2 - b^2| \leq 5 \implies |4 - b^2| \leq 5 \]
This leads to:
\[ -5 \leq 4 - b^2 \leq 5 \]
Rearranging gives:
\[ -1 \leq -b^2 \leq 9 \implies 0 \leq b^2 \leq 9 \]
The values of \( b \) in \( A \) that satisfy this are \( b = 1, 2, 3 \).

- **For \( a = 3 \)**:
\[ |3^2 - b^2| \leq 5 \implies |9 - b^2| \leq 5 \]
This leads to:
\[ -5 \leq 9 - b^2 \leq 5 \]
Rearranging gives:
\[ 4 \leq b^2 \leq 14 \]
The values of \( b \) in \( A \) that satisfy this are \( b = 2, 3 \) (since \( 1^2 = 1 \) which is not ≥ 4).

3. **Constructing the Relation**:
From the calculations:
- For \( a = 1 \): \( b = 1, 2 \) → pairs: \( (1, 1), (1, 2) \)
- For \( a = 2 \): \( b = 1, 2, 3 \) → pairs: \( (2, 1), (2, 2), (2, 3) \)
- For \( a = 3 \): \( b = 2, 3 \) → pairs: \( (3, 2), (3, 3) \)

Thus, the relation \( R \) can be represented as:
\[ R = \{(1, 1), (1, 2), (2, 1), (2, 2), (2, 3), (3, 2), (3, 3)\} \]

4. **Finding Domain and Range**:
- **Domain of \( R \)**: The first elements of the pairs → \( \{1, 2, 3\} \)
- **Range of \( R \)**: The second elements of the pairs → \( \{1, 2, 3\} \)

5. **Identifying the False Statement**:
Now we need to check the given options to find which one is false. Based on the relation we derived:
- The domain is \( \{1, 2, 3\} \).
- The range is \( \{1, 2, 3\} \).
- The relation is symmetric, hence \( R^{-1} = R \).

If one of the statements claims that the range of \( R \) is \( 5 \), that statement is false since the range is actually \( \{1, 2, 3\} \).

### Conclusion:
The false statement is that the range of \( R \) is \( 5 \).