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.