The relation R defined on the set A = {1, 2, 3, 4, 5} by
`R = {(a, b): |a^(2)-b^(2)|lt16}` 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),(4,3),(5,4),(3,4)}`
D. none of these

Explanation

To find the relation \( R \) defined on the set \( A = \{1, 2, 3, 4, 5\} \) by \( R = \{(a, b) : |a^2 - b^2| < 16\} \), we will follow these steps:

### Step 1: Understand the condition
The condition \( |a^2 - b^2| < 16 \) can be rewritten using the difference of squares:
\[ |a^2 - b^2| = |(a - b)(a + b)| < 16 \]
This means that the product of \( (a - b) \) and \( (a + b) \) must be less than 16.

### Step 2: Evaluate for each \( a \) in set \( A \)
We will evaluate \( b \) for each \( a \) in the set \( A \).

#### Case 1: \( a = 1 \)
\[ |1^2 - b^2| < 16 \implies |1 - b^2| < 16 \]
This gives us:
\[ -16 < 1 - b^2 < 16 \]
From \( 1 - b^2 < 16 \):
\[ -b^2 < 15 \implies b^2 > -15 \quad \text{(always true for real } b\text{)} \]
From \( 1 - b^2 > -16 \):
\[ -b^2 > -17 \implies b^2 < 17 \implies -\sqrt{17} < b < \sqrt{17} \]
Since \( b \) must be in \( A \), the possible values of \( b \) are \( 1, 2, 3, 4 \).

#### Case 2: \( a = 2 \)
\[ |2^2 - b^2| < 16 \implies |4 - b^2| < 16 \]
This gives us:
\[ -16 < 4 - b^2 < 16 \]
From \( 4 - b^2 < 16 \):
\[ -b^2 < 12 \implies b^2 > -12 \quad \text{(always true)} \]
From \( 4 - b^2 > -16 \):
\[ -b^2 > -20 \implies b^2 < 20 \implies -\sqrt{20} < b < \sqrt{20} \]
The possible values of \( b \) are \( 1, 2, 3, 4 \).

#### Case 3: \( a = 3 \)
\[ |3^2 - b^2| < 16 \implies |9 - b^2| < 16 \]
This gives us:
\[ -16 < 9 - b^2 < 16 \]
From \( 9 - b^2 < 16 \):
\[ -b^2 < 7 \implies b^2 > -7 \quad \text{(always true)} \]
From \( 9 - b^2 > -16 \):
\[ -b^2 > -25 \implies b^2 < 25 \implies -\sqrt{25} < b < \sqrt{25} \]
The possible values of \( b \) are \( 1, 2, 3, 4, 5 \).

#### Case 4: \( a = 4 \)
\[ |4^2 - b^2| < 16 \implies |16 - b^2| < 16 \]
This gives us:
\[ -16 < 16 - b^2 < 16 \]
From \( 16 - b^2 < 16 \):
\[ -b^2 < 0 \implies b^2 > 0 \implies b \neq 0 \]
From \( 16 - b^2 > -16 \):
\[ -b^2 > -32 \implies b^2 < 32 \implies -\sqrt{32} < b < \sqrt{32} \]
The possible values of \( b \) are \( 1, 2, 3, 4, 5 \).

#### Case 5: \( a = 5 \)
\[ |5^2 - b^2| < 16 \implies |25 - b^2| < 16 \]
This gives us:
\[ -16 < 25 - b^2 < 16 \]
From \( 25 - b^2 < 16 \):
\[ -b^2 < -9 \implies b^2 > 9 \implies b > 3 \]
From \( 25 - b^2 > -16 \):
\[ -b^2 > -41 \implies b^2 < 41 \implies -\sqrt{41} < b < \sqrt{41} \]
The possible values of \( b \) are \( 4, 5 \).

### Step 3: Compile the relation \( R \)
Now we can compile the pairs \( (a, b) \) that satisfy the condition for all values of \( a \):
- For \( a = 1 \): \( (1, 1), (1, 2), (1, 3), (1, 4) \)
- For \( a = 2 \): \( (2, 1), (2, 2), (2, 3), (2, 4) \)
- For \( a = 3 \): \( (3, 1), (3, 2), (3, 3), (3, 4), (3, 5) \)
- For \( a = 4 \): \( (4, 1), (4, 2), (4, 3), (4, 4), (4, 5) \)
- For \( a = 5 \): \( (5, 4), (5, 5) \)

### Final Relation
Thus, the relation \( R \) is:
\[ R = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4), (3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (5, 4), (5, 5)\} \]