To analyze the statement $A \cup B = A \cap C$ and its equivalence to $B \subset A \subset C$, let's break it down step by step.

1. Understanding the Original Equation:

   • $A \cup B$ is the set of all elements that are in either $A$ or $B$ (or both).
   • $A \cap C$ is the set of all elements that are in both $A$ and $C$.

   The equation states that every element in $A$ or $B$ is also in both $A$ and $C$, and vice versa.

2. Interpreting the Implication:

   • The statement $B \subset A$ means that all elements of $B$ are also elements of $A$.
   • The statement $A \subset C$ means that all elements of $A$ are also elements of $C$.

3. Proof of Equivalence:

   • (If): Assume $B \subset A$ and $A \subset C$.

     • Then, every element of $B$ is in $A$ and hence also in $C$ (since $A \subset C$).
     • Thus, $A \cup B = A$ (since $B$ is contained in $A$).
     • We have $A = A \cap C$, which holds true as every element in $A$ is also in $C$.

   • (Only If): Assume $A \cup B = A \cap C$.

     • Then, all elements of $B$ must be in $A$ (otherwise, $A \cup B$ would contain elements not in $A$).
     • Moreover, since $A \cup B$ is equal to $A \cap C$, it follows that all elements of $A$ must be in $C$ as well.

Thus, the statement $A \cup B = A \cap C$ is equivalent to $B \subset A \subset C$.

Summary

The original equation can be rewritten as the implication that $B$ is a subset of $A$, and $A$ is a subset of $C$.

If you need any further details or have questions, feel free to ask! Here are some related questions you might consider:

1. What are the definitions of union and intersection in set theory?
2. Can you provide examples of sets $A$, $B$, and $C$ that satisfy this equivalence?
3. How would this change if we removed the subset conditions?
4. What happens if $A$ and $C$ are equal sets?
5. How can we visualize this relationship with Venn diagrams?

Tip: When working with set relations, drawing a Venn diagram can often clarify how sets overlap and relate to each other.