https://file.aiquickdraw.com/mathbot/file/1726163916130vm28v8pv.jpg

The statement in the image reads:

"If \( A \subseteq B \), then \( n(A) \leq n(B) \)."

This statement is true. Here’s why:

1. **Subset Definition**: If \( A \) is a subset of \( B \) (\( A \subseteq B \)), every element of \( A \) is also an element of \( B \).
2. **Cardinality**: The notation \( n(A) \) represents the cardinality (or number of elements) of set \( A \).
3. **Implication**: Since \( A \) is a subset of \( B \), the number of elements in \( A \) cannot exceed the number of elements in \( B \). This means that \( n(A) \leq n(B) \).

Therefore, the statement is **true**.

Would you like further details or have any questions?

Here are some related questions to expand your understanding:

1. What is the definition of a proper subset?
2. How does the cardinality of the union of two sets \( A \) and \( B \) relate to the cardinalities of \( A \) and \( B \)?
3. What happens to the cardinality relationship if \( A \) is a proper subset of \( B \) (\( A \subset B \))?
4. Can you provide an example where \( A \subseteq B \) and illustrate \( n(A) \leq n(B) \) with specific numbers?
5. How does the inclusion-exclusion principle apply to the cardinalities of sets?

**Tip:** When dealing with set operations and cardinality, visualizing with Venn diagrams can often help clarify relationships and make understanding easier.

Explore the relationship between subsets and cardinality. Learn why if A is a subset of B (A ⊆ B), then the number of elements in A (n(A)) is less than or equal to the number of elements in B (n(B)). This is a key concept in set theory, suitable for high school students.

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