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., $A \neq B$).