Let `R` be set of points inside a rectangle of sides `a` and `b` `(a, b > 1)` with two sides along the positive direction of x-axis and y-axis(a) `R = {(x, y) : 0 le x le a, 0 le y le b}`(b) `R = {(x,y) : 0 le x lt a, 0 le y le b}`(c) `R = {(x,y): 0 le x le a, 0 lt y lt b}`(d) `R = {(x,y): 0 lt x lt a, 0 lt y lt b}`

Options

A. `R = {(x, y) : 0 le x le a, 0 le y le b}`
B. `R = {(x,y) : 0 le x lt a, 0 le y le b}`
C. `R = {(x,y): 0 le x le a, 0 lt y lt b}`
D. `R = {(x,y): 0 lt x lt a, 0 lt y lt b}`

Explanation

To solve the problem, we need to analyze the set of points inside a rectangle defined by the sides \( a \) and \( b \) in the first quadrant of the Cartesian plane. The rectangle has its sides along the positive x-axis and y-axis.

### Step-by-step Solution:

1. **Understanding the Rectangle**:
The rectangle is defined by the points (0, 0) at the origin, (a, 0) on the x-axis, (0, b) on the y-axis, and (a, b) at the opposite corner. The rectangle lies entirely in the first quadrant since both \( a \) and \( b \) are greater than 1.

2. **Defining the Set of Points**:
The set \( R \) consists of all points \( (x, y) \) that lie within this rectangle. We need to determine the conditions for \( x \) and \( y \) based on the boundaries of the rectangle.

3. **Conditions for \( x \)**:
- Since the rectangle extends from 0 to \( a \) along the x-axis, the possible values for \( x \) must satisfy:
\[ 0 \leq x \leq a \]
- However, since we are looking for points **inside** the rectangle, \( x \) cannot equal \( a \). Thus, we have:
\[ 0 \leq x < a \]

4. **Conditions for \( y \)**:
- Similarly, for the y-coordinate, the rectangle extends from 0 to \( b \) along the y-axis. The possible values for \( y \) must satisfy:
\[ 0 \leq y \leq b \]
- Again, since we are looking for points **inside** the rectangle, \( y \) cannot equal \( b \). Thus, we have:
\[ 0 \leq y < b \]

5. **Combining the Conditions**:
- Combining the conditions for both \( x \) and \( y \), we can express the set \( R \) as:
\[ R = \{(x, y) : 0 < x < a, \, 0 < y < b\} \]

6. **Identifying the Correct Option**:
- Now, we can compare our derived set with the given options:
- (a) \( R = \{(x, y) : 0 \leq x \leq a, \, 0 \leq y \leq b\} \) - includes the boundaries, not correct.
- (b) \( R = \{(x, y) : 0 \leq x < a, \, 0 \leq y \leq b\} \) - includes the boundary for \( y \), not correct.
- (c) \( R = \{(x, y) : 0 \leq x \leq a, \, 0 < y < b\} \) - includes the boundary for \( x \), not correct.
- (d) \( R = \{(x, y) : 0 < x < a, \, 0 < y < b\} \) - matches our derived set, correct.

### Conclusion:
The correct option is (d) \( R = \{(x, y) : 0 < x < a, \, 0 < y < b\} \).