The statement \( A \subset B \iff A \cap B = A \) is true and can be explained as follows:

### Explanation:

- \( A \subset B \) means that **A is a subset of B**, or every element of A is also an element of B.
  
- \( A \cap B \) represents the **intersection** of sets A and B, meaning it includes only those elements that are in both A and B.

Thus, when \( A \subset B \), every element of A is also in B, which implies that:

\[
A \cap B = A
\]

because all elements of A are in both A and B, so the intersection would just be A itself. Therefore, the statement \( A \subset B \iff A \cap B = A \) holds true.

### Recap of the equivalence:

- **\( A \subset B \)** means every element of A is in B.
- **\( A \cap B = A \)** means that the intersection of A and B contains all the elements of A (and possibly others from B).

Thus, both expressions are logically equivalent.

Would you like more details or have any questions?

### Related Questions:
1. What is the difference between \( A \subseteq B \) and \( A \subset B \)?
2. Can \( A \cap B = A \) if \( A \) is not a subset of \( B \)?
3. How is the intersection \( A \cap B \) defined for disjoint sets?
4. What is the relationship between \( A \subset B \) and \( A \cup B \)?
5. What are some examples where \( A \cap B = \emptyset \)?

### Tip:
Remember, when \( A \subset B \), this doesn't necessarily mean \( A = B \); it simply means all elements of A are contained in B.

Learn about the equivalence between subsets and intersections with the statement A ⊂ B ⇔ A ∩ B = A. This set theory problem explains how the intersection of sets A and B relates to subset relationships, providing a clear mathematical explanation. Suitable for students in Grades 9-12.

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