To solve the problem, we need to evaluate the two statements regarding the sets A and B and their symmetric difference, denoted as \( A \triangle B \).
### Step-by-Step Solution:
**Step 1: Define the Sets A and B**
- Given:
- \( A = \{ x \mid x \in \mathbb{R}, x \geq 2 \} \)
- \( B = \{ x \mid x \in \mathbb{R}, x < 4 \} \)
**Step 2: Express Sets A and B in Interval Notation**
- Set A can be expressed as:
- \( A = [2, \infty) \)
- Set B can be expressed as:
- \( B = (-\infty, 4) \)
**Step 3: Find the Symmetric Difference \( A \triangle B \)**
- The symmetric difference \( A \triangle B \) is defined as:
- \( A \triangle B = (A - B) \cup (B - A) \)
**Step 4: Calculate \( A - B \)**
- \( A - B \) consists of elements in A that are not in B:
- Since \( A = [2, \infty) \) and \( B = (-\infty, 4) \), the elements of A that are not in B are those greater than or equal to 4.
- Thus, \( A - B = [4, \infty) \)
**Step 5: Calculate \( B - A \)**
- \( B - A \) consists of elements in B that are not in A:
- The elements of B that are not in A are those less than 2.
- Thus, \( B - A = (-\infty, 2) \)
**Step 6: Combine the Results**
- Now, we combine the results from steps 4 and 5:
- \( A \triangle B = (A - B) \cup (B - A) = [4, \infty) \cup (-\infty, 2) \)
**Step 7: Express the Result in Set Notation**
- The union of these two intervals can be expressed as:
- \( A \triangle B = (-\infty, 2) \cup [4, \infty) \)
**Step 8: Compare with the Given Statement**
- The statement claims that \( A \triangle B = \mathbb{R} - [2, 4) \).
- The set \( \mathbb{R} - [2, 4) \) is indeed equal to \( (-\infty, 2) \cup [4, \infty) \).
### Conclusion:
- Both statements are true:
- Statement 1 is true: \( A \triangle B = \mathbb{R} - [2, 4) \)
- Statement 2 is true: The definition of symmetric difference is correct.
