use venn diagrams to prove illustrate the relatinship A ⊆ B and B ⊆ C

To illustrate the relationship \( A \subseteq B \) and \( B \subseteq C \) using Venn diagrams, we can represent each of the sets \( A \), \( B \), and \( C \) as overlapping circles, where:

- \( A \subseteq B \) means that all elements of \( A \) are contained within \( B \).
- \( B \subseteq C \) means that all elements of \( B \) are contained within \( C \).

### Step-by-Step Construction of the Venn Diagram:

1. **Circle for Set \( C \):**
   - Draw a large circle to represent set \( C \). This is the largest set in the relationship since \( B \subseteq C \).

2. **Circle for Set \( B \):**
   - Inside the circle for \( C \), draw a smaller circle to represent set \( B \), since \( B \subseteq C \).

3. **Circle for Set \( A \):**
   - Inside the circle for \( B \), draw another smaller circle to represent set \( A \), since \( A \subseteq B \).

### Interpretation:
- The smallest circle (representing \( A \)) is completely within the next circle (representing \( B \)).
- The circle for \( B \) is completely within the largest circle (representing \( C \)).

This visual structure directly shows the relationships:
- \( A \subseteq B \) because all elements of \( A \) are within \( B \).
- \( B \subseteq C \) because all elements of \( B \) are within \( C \).
- Hence, \( A \subseteq C \) as well by transitivity.

Would you like a detailed image of this Venn diagram?

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### 5 Related Questions:
1. What is the difference between \( A \subseteq B \) and \( A \subset B \)?
2. How would you represent the relationship if \( A \subseteq B \) but \( A \neq B \)?
3. Can Venn diagrams illustrate disjoint sets? How?
4. What happens if \( A \) and \( B \) are overlapping but neither is a subset of the other?
5. How would you modify the Venn diagram if \( B = C \)?

### Tip:
Always remember that in subset notation, \( A \subseteq B \) allows \( A \) and \( B \) to be equal, while \( A \subset B \) implies strict containment (i.e., \( A \neq B \)).

use venn diagrams to prove illustrate the relationship A ⊆ B and B ⊆ C

Explore how to use Venn diagrams to visually prove the relationship A ⊆ B and B ⊆ C. Understand the concept of subsets and transitivity in set theory with this step-by-step guide, perfect for students in Grades 9-12.

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