Let `A={x : x in R,-1 lt x lt 1}`, `B={x : x in R, x le0 or x ge 2}` and `AuuB=R-D`, then find set `D`

Answer

To solve the problem, we need to find the set \( D \) given the sets \( A \) and \( B \) and the equation \( A \cup B = \mathbb{R} - D \).

### Step-by-step Solution:

1. **Define the Sets**:
- Set \( A = \{ x \in \mathbb{R} : -1 < x < 1 \} \)
- Set \( B = \{ x \in \mathbb{R} : x \leq 0 \text{ or } x \geq 2 \} \)

2. **Visualize the Sets**:
- Set \( A \) includes all real numbers between -1 and 1 (not including -1 and 1).
- Set \( B \) includes all real numbers that are either less than or equal to 0 or greater than or equal to 2.

3. **Determine the Union \( A \cup B \)**:
- The union \( A \cup B \) will include:
- All values from set \( A \) which are between -1 and 1.
- All values from set \( B \) which includes values less than or equal to 0 and values greater than or equal to 2.
- Therefore, \( A \cup B \) can be visualized as:
- From \( -1 \) to \( 1 \) (from set \( A \))
- Plus all values less than or equal to 0 (from set \( B \))
- Plus all values greater than or equal to 2 (from set \( B \))

4. **Identify the Range of \( A \cup B \)**:
- The union \( A \cup B \) covers:
- All values from \( -1 \) to \( 0 \) (from \( A \) and part of \( B \))
- All values from \( 2 \) to \( \infty \) (from \( B \))
- Thus, \( A \cup B = (-1, 0] \cup [2, \infty) \)

5. **Determine \( D \)**:
- According to the problem, \( A \cup B = \mathbb{R} - D \).
- This means \( D \) consists of the values that are not included in \( A \cup B \).
- The only intervals not covered by \( A \cup B \) are:
- The interval \( (0, 2) \)
- Therefore, we can conclude that:
- \( D = (0, 2) \)

### Final Answer:
The set \( D \) is \( (0, 2) \).