The relation R defined on the set A = {1,2,3,4, 5} by
R ={(x, y)} : `|x^(2) -y^(2) | lt 16 }` is given by

Options

A. {(1,1), (2, 1), (3, 1),(4, 1), (2,3)}
B. {(2,2), (3,2),(4,2), (2,4)}
C. {(3,3), (3,4), (5,4), (4, 3), (3, 1)}
D. None of these

Answer

To find the relation \( R \) defined on the set \( A = \{1, 2, 3, 4, 5\} \) by the condition \( |x^2 - y^2| < 16 \), we will systematically evaluate all possible ordered pairs \( (x, y) \) where \( x, y \in A \).

### Step-by-Step Solution:

1. **Identify the Set**:
The set \( A \) is given as \( A = \{1, 2, 3, 4, 5\} \).

2. **Understand the Condition**:
The relation \( R \) consists of ordered pairs \( (x, y) \) such that \( |x^2 - y^2| < 16 \). This can be rewritten using the difference of squares:
\[ |x^2 - y^2| = |(x - y)(x + y)| \]
Thus, we need to find pairs \( (x, y) \) for which this product is less than 16.

3. **Generate Ordered Pairs**:
We will evaluate each possible pair \( (x, y) \) from the set \( A \).

4. **Evaluate Each Pair**:
- For \( x = 1 \):
- \( (1, 1) \): \( |1^2 - 1^2| = |0| < 16 \) → Include
- \( (1, 2) \): \( |1^2 - 2^2| = |1 - 4| = |3| < 16 \) → Include
- \( (1, 3) \): \( |1^2 - 3^2| = |1 - 9| = |8| < 16 \) → Include
- \( (1, 4) \): \( |1^2 - 4^2| = |1 - 16| = |15| < 16 \) → Include
- \( (1, 5) \): \( |1^2 - 5^2| = |1 - 25| = |24| \not< 16 \) → Exclude

- For \( x = 2 \):
- \( (2, 1) \): \( |2^2 - 1^2| = |4 - 1| = |3| < 16 \) → Include
- \( (2, 2) \): \( |2^2 - 2^2| = |0| < 16 \) → Include
- \( (2, 3) \): \( |2^2 - 3^2| = |4 - 9| = |5| < 16 \) → Include
- \( (2, 4) \): \( |2^2 - 4^2| = |4 - 16| = |12| < 16 \) → Include
- \( (2, 5) \): \( |2^2 - 5^2| = |4 - 25| = |21| \not< 16 \) → Exclude

- For \( x = 3 \):
- \( (3, 1) \): \( |3^2 - 1^2| = |9 - 1| = |8| < 16 \) → Include
- \( (3, 2) \): \( |3^2 - 2^2| = |9 - 4| = |5| < 16 \) → Include
- \( (3, 3) \): \( |3^2 - 3^2| = |0| < 16 \) → Include
- \( (3, 4) \): \( |3^2 - 4^2| = |9 - 16| = |7| < 16 \) → Include
- \( (3, 5) \): \( |3^2 - 5^2| = |9 - 25| = |16| \not< 16 \) → Exclude

- For \( x = 4 \):
- \( (4, 1) \): \( |4^2 - 1^2| = |16 - 1| = |15| < 16 \) → Include
- \( (4, 2) \): \( |4^2 - 2^2| = |16 - 4| = |12| < 16 \) → Include
- \( (4, 3) \): \( |4^2 - 3^2| = |16 - 9| = |7| < 16 \) → Include
- \( (4, 4) \): \( |4^2 - 4^2| = |0| < 16 \) → Include
- \( (4, 5) \): \( |4^2 - 5^2| = |16 - 25| = |9| < 16 \) → Include

- For \( x = 5 \):
- \( (5, 1) \): \( |5^2 - 1^2| = |25 - 1| = |24| \not< 16 \) → Exclude
- \( (5, 2) \): \( |5^2 - 2^2| = |25 - 4| = |21| \not< 16 \) → Exclude
- \( (5, 3) \): \( |5^2 - 3^2| = |25 - 9| = |16| \not< 16 \) → Exclude
- \( (5, 4) \): \( |5^2 - 4^2| = |25 - 16| = |9| < 16 \) → Include
- \( (5, 5) \): \( |5^2 - 5^2| = |0| < 16 \) → Include

5. **Compile the Results**:
After evaluating all pairs, we find the relation \( R \):
\[ R = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5, 4), (5, 5)\} \]