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The question here asks for a counterexample to the statement:

\[
A \subseteq B \iff A \cup B = A
\]

To find a counterexample, we need to show that this statement does not hold universally. The statement essentially implies that if \( A \) is a subset of \( B \), then the union of \( A \) and \( B \) should equal \( A \), and vice versa.

### Explanation
1. **If \( A \subseteq B \)**, then every element in \( A \) is also in \( B \). However, this does not imply that \( A \cup B = A \); in fact, \( A \cup B \) would typically be equal to \( B \) if \( A \subseteq B \) and \( B \neq A \).
   
2. The statement would be true only if \( A = B \) because only then would \( A \cup B = A \).

### Counterexample
Let’s choose sets such that \( A \subseteq B \) but \( A \cup B \neq A \). Consider:

- \( A = \{1\} \)
- \( B = \{1, 2\} \)

Here:
- \( A \subseteq B \) is true since \( \{1\} \) is indeed a subset of \( \{1, 2\} \).
- However, \( A \cup B = \{1, 2\} \), which is not equal to \( A = \{1\} \).

Thus, \( A \subseteq B \) does not imply that \( A \cup B = A \) in this case, providing a counterexample to the statement.

Would you like further details on this solution, or have any other questions?

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Here are 5 related questions for deeper exploration:
1. What is the difference between a subset and a proper subset?
2. How does the union of two sets behave when one set is a subset of another?
3. Can you provide an example where \( A \cup B = A \) and \( A = B \)?
4. What conditions are necessary for \( A \cup B = A \)?
5. How does the intersection of sets relate to their union?

**Tip:** When dealing with set relationships, visualize with simple examples or Venn diagrams to better understand subset and union properties.

Find a counterexample to A ⊆ B ⇔ A ∪ B = A.

Explore a counterexample to the set theory statement A ⊆ B ⇔ A ∪ B = A. Understand the relationships between subsets and unions with a specific example showing where this equivalence fails. Suitable for high school students studying set theory.

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