Find counter-example: if A and B are sets, then |A+B|=|A|+|A|

The statement \(|A + B| = |A| + |B|\) is incorrect, and we can find a counterexample by interpreting "A + B" in different ways, such as the union of two sets. The correct interpretation of the equation should be clarified. However, assuming \(A + B\) refers to the union of the sets \(A\) and \(B\), the correct relation would be:

\[
|A \cup B| = |A| + |B| - |A \cap B|
\]

This formula accounts for the overlap between sets \(A\) and \(B\). So, the given equation \(|A + B| = |A| + |B|\) is only true when \(A\) and \(B\) are disjoint (i.e., \(A \cap B = \emptyset\)).

### Counterexample:
Let:
- \(A = \{1, 2\}\)
- \(B = \{2, 3\}\)

Now,
- \(|A| = 2\)
- \(|B| = 2\)
- \(A \cup B = \{1, 2, 3\}\), so \(|A \cup B| = 3\).

Clearly, \(|A + B| = 3\) and \(|A| + |B| = 4\). Hence, \(|A + B| \neq |A| + |B|\), providing a counterexample to the given statement.

### Would you like more details on how we find intersections and unions of sets?

Here are 5 questions to deepen your understanding:
1. How does the formula for the union of two sets change when they overlap?
2. Can we extend this reasoning to more than two sets? How?
3. What happens when \(A\) and \(B\) are subsets of each other?
4. How can Venn diagrams help visualize these concepts?
5. How would the result change if we took the Cartesian product instead of the union?

**Tip:** Always check for overlaps between sets when calculating their union, as it impacts the total number of distinct elements.

Explore a counter-example to the statement |A+B| = |A| + |B| using set theory. Understand the correct formula |A ∪ B| = |A| + |B| - |A ∩ B| and learn how to handle overlapping sets. Suitable for students in Grades 9-12.

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