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The statement provided is:

"For two finite sets \( A \) and \( B \): 
\[
A \neq B \implies \#(A \cap B) < \#B
\]
"

Where:
- \( A \neq B \) means that sets \( A \) and \( B \) are not equal.
- \( \#(A \cap B) \) refers to the cardinality (number of elements) of the intersection of \( A \) and \( B \).
- \( \#B \) is the cardinality of set \( B \).

### Analysis:
The statement claims that if \( A \) and \( B \) are different sets, then the number of elements in \( A \cap B \) (the intersection) must be strictly less than the number of elements in \( B \).

However, this is not necessarily true. Consider the case where:
- \( B \subseteq A \), meaning all elements of \( B \) are contained in \( A \), but \( A \) has additional elements not in \( B \). In this case, \( A \neq B \), but \( A \cap B = B \), so the cardinality \( \#(A \cap B) \) would equal \( \#B \), not be less than it.

This provides a counterexample, showing the statement is **false**. The cardinality of the intersection is not always strictly less than that of \( B \), even if \( A \neq B \).

Thus, the selected answer "False" is correct.

Would you like more details or have any further questions?

Here are 5 related questions:
1. What is the definition of set equality?
2. How can we find the intersection of two sets?
3. Can the intersection of two sets ever be larger than one of the sets?
4. What happens when one set is a subset of the other?
5. What is the union of two sets, and how does it differ from the intersection?

**Tip:** When proving or disproving set relations, always look for counterexamples to test the general validity of the statement.

For two finite sets A, B: A ≠ B ⇒ #(A ∩ B) < #B

Explore the relationship between set intersections and cardinality with the example of two finite sets A and B. This problem challenges the idea that the intersection of two unequal sets has a cardinality strictly less than that of B. Suitable for Grades 9-12 students studying set theory.

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