To solve the problem step by step, we will first define the sets A and B in roster form, and then find their union, intersection, and differences.
### Step 1: Define Set A in Roster Form
Set A is defined as:
\[ A = \{ x : x \in \mathbb{N} \text{ and } 3 < x \leq 6 \} \]
The natural numbers (N) greater than 3 and less than or equal to 6 are:
- 4
- 5
- 6
Thus, in roster form:
\[ A = \{ 4, 5, 6 \} \]
### Step 2: Define Set B in Roster Form
Set B is defined as:
\[ B = \{ x : x \in \mathbb{W} \text{ and } x < 4 \} \]
The whole numbers (W) less than 4 are:
- 0
- 1
- 2
- 3
Thus, in roster form:
\[ B = \{ 0, 1, 2, 3 \} \]
### Step 3: Find A ∪ B (Union of A and B)
The union of two sets A and B includes all elements that are in A, in B, or in both:
\[ A \cup B = \{ 4, 5, 6 \} \cup \{ 0, 1, 2, 3 \} \]
Combining the elements, we get:
\[ A \cup B = \{ 0, 1, 2, 3, 4, 5, 6 \} \]
### Step 4: Find A ∩ B (Intersection of A and B)
The intersection of two sets A and B includes only the elements that are in both A and B:
\[ A \cap B = \{ 4, 5, 6 \} \cap \{ 0, 1, 2, 3 \} \]
Since there are no common elements between A and B, we have:
\[ A \cap B = \emptyset \]
### Step 5: Find A - B (Difference of A and B)
The difference A - B includes the elements that are in A but not in B:
\[ A - B = \{ 4, 5, 6 \} - \{ 0, 1, 2, 3 \} \]
Since none of the elements of A are in B, we have:
\[ A - B = \{ 4, 5, 6 \} \]
### Step 6: Find B - A (Difference of B and A)
The difference B - A includes the elements that are in B but not in A:
\[ B - A = \{ 0, 1, 2, 3 \} - \{ 4, 5, 6 \} \]
Again, since none of the elements of B are in A, we have:
\[ B - A = \{ 0, 1, 2, 3 \} \]
### Final Answers
1. \( A = \{ 4, 5, 6 \} \)
2. \( B = \{ 0, 1, 2, 3 \} \)
3. \( A \cup B = \{ 0, 1, 2, 3, 4, 5, 6 \} \)
4. \( A \cap B = \emptyset \)
5. \( A - B = \{ 4, 5, 6 \} \)
6. \( B - A = \{ 0, 1, 2, 3 \} \)
