If `A = {x in R : 0 lt x lt 4} and B = {x in R : 1 le x le 7}` then find `A Delta B`.

Answer

To find the asymmetric difference \( A \Delta B \) of the sets \( A \) and \( B \), we will follow these steps:

### Step 1: Define the Sets
Given:
- \( A = \{ x \in \mathbb{R} : 0 < x < 4 \} \)
- \( B = \{ x \in \mathbb{R} : 1 \leq x \leq 7 \} \)

### Step 2: Identify the Intervals
From the definitions:
- Set \( A \) includes all real numbers between 0 and 4 (not including 0 and 4).
- Set \( B \) includes all real numbers from 1 to 7 (including 1 and 7).

### Step 3: Visualize the Sets on a Number Line
On a number line:
- \( A \) is represented as an open interval (0, 4).
- \( B \) is represented as a closed interval [1, 7].

### Step 4: Find \( A - B \)
The set \( A - B \) contains elements in \( A \) that are not in \( B \):
- The overlap between \( A \) and \( B \) is from 1 to 4 (since \( A \) goes up to 4 and \( B \) starts from 1).
- Therefore, \( A - B \) will include elements from \( A \) that are less than 1.

Thus,
\[ A - B = \{ x \in \mathbb{R} : 0 < x < 1 \} \]

### Step 5: Find \( B - A \)
The set \( B - A \) contains elements in \( B \) that are not in \( A \):
- The overlap is again from 1 to 4.
- Therefore, \( B - A \) will include elements from \( B \) that are greater than 4.

Thus,
\[ B - A = \{ x \in \mathbb{R} : 4 < x \leq 7 \} \]

### Step 6: Combine the Results
The asymmetric difference \( A \Delta B \) is defined as:
\[ A \Delta B = (A - B) \cup (B - A) \]
Substituting the results from Steps 4 and 5:
\[ A \Delta B = \{ x \in \mathbb{R} : 0 < x < 1 \} \cup \{ x \in \mathbb{R} : 4 < x \leq 7 \} \]

### Final Result
Thus, the asymmetric difference \( A \Delta B \) can be expressed as:
\[ A \Delta B = \{ x \in \mathbb{R} : 0 < x < 1 \text{ or } 4 < x \leq 7 \} \]