https://file.aiquickdraw.com/mathbot/file/1726097872259zbkrqdnx.jpg

Let's analyze the sets provided in the image:

1. **Set A**: \( A = \{x \in \mathbb{Q} : 0 < x < 20\} \), which is the set of rational numbers between 0 and 20.
2. **Set B**: \( B = \{4, 8, 12, 16\} \), a set with specific integers.
3. **Set C**: \( C = \{x \in \mathbb{Z} : x \text{ is an integer multiple of } 4\} \), which includes multiples of 4.
4. **Set D**: \( D = \{0, 4, 8, 12, 16, 20\} \), a set with even numbers between 0 and 20.
5. **Set E**: \( E = \{x \in \mathbb{N} : 0 \leq x \leq 20\} \), which represents the set of natural numbers between 0 and 20.
6. **Set F**: \( F = \{4, 8, 12, 16, 0, 20\} \), which is identical to Set D.

### Now, let's evaluate the subset relationships:

1. **\( D \subseteq C \)**:
   - \( D = \{0, 4, 8, 12, 16, 20\} \)
   - \( C \) includes multiples of 4, which are \( \{0, 4, 8, 12, 16, 20, \ldots\} \).
   - **True**, because every element in \( D \) is also in \( C \).

2. **\( B \subseteq D \)**:
   - \( B = \{4, 8, 12, 16\} \)
   - \( D = \{0, 4, 8, 12, 16, 20\} \)
   - **True**, because all elements of \( B \) are in \( D \).

3. **\( E \subseteq D \)**:
   - \( E = \{0, 1, 2, 3, 4, 5, 6, \ldots, 20\} \)
   - \( D = \{0, 4, 8, 12, 16, 20\} \)
   - **False**, because \( E \) contains numbers (like 1, 2, etc.) that are not in \( D \).

4. **\( A \subseteq B \)**:
   - \( A \) is the set of all rational numbers between 0 and 20.
   - \( B = \{4, 8, 12, 16\} \)
   - **False**, because \( A \) contains many rational numbers not in \( B \).

### Final Answers:
- \( D \subseteq C \) is **true**.
- \( B \subseteq D \) is **true**.
- \( E \subseteq D \) is **false**.
- \( A \subseteq B \) is **false**.

Would you like more details or have any questions?

Here are 5 related questions to consider:
1. What is the difference between rational numbers and integers?
2. How can you determine whether a set is a subset of another?
3. Why is \( A \) not a subset of \( B \), even though both contain numbers between 0 and 20?
4. Can a set be a subset of itself?
5. How would these relationships change if the sets included different ranges?

**Tip**: A subset relation holds if and only if every element of the first set is also an element of the second set.

Use the definitions for the sets given below to determine whether each statement is true or false:

A = {x ∈ Q: 0 < x < 20}
B = {4, 8, 12, 16}
C = {x ∈ Z: x is an integer multiple of 4}
D = {0, 4, 8, 12, 16, 20}
E = {x ∈ N: 0 ≤ x ≤ 20}
F = {4, 8, 12, 16, 0, 20}

Select the statements that are true:

1. D ⊆ C
2. B ⊆ D
3. E ⊆ D
4. A ⊆ B

Analyze the subset relationships between given sets A, B, C, D, E, and F to determine which statements are true or false. This problem covers key concepts in set theory, including subsets and mathematical logic, suitable for students in Grades 8-10.

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