Let R be the set of points inside a rectangle of sides a and b `(a,b gt1)` with two sides along the positive direction of x-axis and y-axis. Then
(i) `R={(x,y):0lexlea,aleyleb}`
(ii) `R={(2,y):0lexlta,0leyleb}`
(iii) `R={(x,y):0lexlea,0ltyltb}`
(iv) `R'={(x,y):0ltxlta,0ltyltb}`

Options

A. `R={(x,y):0lexlea,aleyleb}`
B. `R={(2,y):0lexlta,0leyleb}`
C. `R={(x,y):0lexlea,0ltyltb}`
D. `R'={(x,y):0ltxlta,0ltyltb}`

Answer

To solve the problem, we need to determine the set of points \( R \) that lie inside a rectangle defined by its sides \( a \) and \( b \) (where \( a, b > 1 \)) in the first quadrant. The rectangle has its sides aligned with the positive directions of the x-axis and y-axis.

### Step-by-Step Solution:

1. **Understanding the Rectangle**:
- The rectangle is defined in the first quadrant with one vertex at the origin (0,0).
- The other vertices of the rectangle are at (a, 0), (0, b), and (a, b).

2. **Identifying the Points Inside the Rectangle**:
- Since we are looking for points inside the rectangle, we need to consider the boundaries of the rectangle.
- The x-coordinates of points inside the rectangle must satisfy \( 0 < x < a \).
- The y-coordinates of points inside the rectangle must satisfy \( 0 < y < b \).

3. **Formulating the Set \( R \)**:
- Based on the conditions derived, we can express the set \( R \) as:
\[ R = \{(x, y) : 0 < x < a, 0 < y < b\} \]
- This means that \( x \) can take any value between 0 and \( a \) (exclusive), and \( y \) can take any value between 0 and \( b \) (exclusive).

4. **Identifying the Correct Option**:
- Now, we compare our derived set \( R \) with the given options:
- (i) \( R = \{(x,y) : 0 \leq x \leq a, a \leq y \leq b\} \) - Incorrect, as it includes boundaries and has incorrect y-range.
- (ii) \( R = \{(2,y) : 0 \leq x < a, 0 \leq y \leq b\} \) - Incorrect, as it fixes \( x \) at 2.
- (iii) \( R = \{(x,y) : 0 \leq x \leq a, 0 < y < b\} \) - Incorrect, as it includes the boundary for \( x \).
- (iv) \( R' = \{(x,y) : 0 < x < a, 0 < y < b\} \) - Correct, as it matches our derived set.

5. **Conclusion**:
- The correct option is (iv) \( R' = \{(x,y) : 0 < x < a, 0 < y < b\} \).