https://file.aiquickdraw.com/mathbot/file/1725801887155hq0ubhtv.jpg

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!

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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 \( \in \).

Solve the set theory problems involving subsets, elements, and natural numbers.

Explore how to solve set theory problems involving subsets, element membership, and the classification of natural numbers. This problem covers key concepts in set theory, including subsets, elements, and the rules governing natural numbers. Suitable for students in Grades 6-8.

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