I've opened the image with the mathematical problems. I will now solve the problems step by step.

Let's go through them one by one:

Part I: True or False Statements

Given $A = \{0, 1, 2, 3, 4, 5, 6\}$:

1. $2 \in A$
   • True, because 2 is an element of set $A$.
2. $8 \in A$
   • False, 8 is not in set $A$.
3. $1 \subseteq A$
   • False, 1 is an element, not a subset.
4. $\{1, 2, 3\} \subseteq A$
   • True, because all elements 1, 2, and 3 are in $A$.
5. $\{0, 8, 2\} \subseteq A$
   • False, because 8 is not in $A$.
6. $\{1, 7\} \subseteq A$
   • False, because 7 is not in $A$.
7. $\emptyset \subseteq A$
   • True, the empty set is a subset of any set.
8. $\{1, 2, 6\} \subseteq A$
   • True, because all elements 1, 2, and 6 are in $A$.

Part II: Subsets of the Given Set $B = \{a, b, c\}$

1. $\emptyset$ is a subset of $B$.
2. $\{a\}$
3. $\{b\}$
4. $\{c\}$
5. $\{a, b\}$
6. $\{a, c\}$
7. $\{b, c\}$
8. $\{a, b, c\}$

Part III: Inserting $\in$ or $\notin$

Given sets:

• $C = \{Red, Blue, Yellow\}$
• $D = \{0, 1, 3, 5, 7\}$
• $Z = \{2, 4, 6\}$

1. $8 \notin D$
2. $Red \in C$
3. $Green \notin C$
4. $2 \in Z$
5. $3 \notin Z$
6. $Yellow \in C$

Part IV: Explanation of Set Terminology

1. The set of natural numbers:

   • The set of natural numbers consists of all positive integers starting from 1, denoted as $\{1, 2, 3, \dots\}$.

2. The set of integers:

   • The set of integers consists of all positive and negative whole numbers, including zero, denoted as $\{ \dots, -3, -2, -1, 0, 1, 2, 3, \dots\}$.

3. The set of real numbers:

   • The set of real numbers includes all rational and irrational numbers, which covers integers, fractions, and non-repeating, non-terminating decimals.

4. The set of rational numbers:

   • The set of rational numbers includes any number that can be expressed as the quotient $\frac{a}{b}$, where $a$ and $b$ are integers, and $b \neq 0$.

If you have any questions or want more details on any part of this solution, feel free to ask!

Here are 5 related questions you can try:

1. Can the empty set be a subset of every set?
2. What is the difference between $\in$ and $\subseteq$?
3. Is $\{1, 2, 3\} \subseteq \{1, 2, 3, 4\}$?
4. What is the cardinality of the set $\{a, b, c, d\}$?
5. What is the difference between rational and irrational numbers?

Tip: A subset contains all its elements within another set, but not every element needs to be present for $\in$.