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true or false. give a counter example if false

The statement provided is:

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

Here:
- \( n(A) \) represents the cardinality (number of elements) of set \( A \),
- \( n(B) \) represents the cardinality of set \( B \),
- \( A \subseteq B \) means that set \( A \) is a subset of set \( B \) (i.e., all elements of \( A \) are also in \( B \)).

### Let's analyze the statement:

- The statement says that if the number of elements in set \( A \) is less than the number of elements in set \( B \), then \( A \) must be a subset of \( B \).
  
This is **false** because a set can have fewer elements than another set without being a subset of that set.

### Counterexample:
Let:
- \( A = \{1, 2\} \)
- \( B = \{3, 4, 5\} \)

In this case:
- \( n(A) = 2 \) and \( n(B) = 3 \), so \( n(A) < n(B) \),
- However, \( A \subseteq B \) is **false** because the elements of \( A \) are not contained in \( B \).

Thus, this provides a counterexample, disproving the statement.

Would you like further details or clarification?

### Related Questions:
1. What is the difference between a proper subset and a subset?
2. How is the cardinality of a set determined?
3. Can a set with a greater cardinality ever be a subset of a set with a smaller cardinality?
4. What is the difference between an element of a set and a subset of a set?
5. How do the rules of set inclusion apply in infinite sets?

### Tip:
A subset relationship implies inclusion of elements, while cardinality only concerns the count of elements—be cautious not to confuse them!

If n(A) < n(B), then A ⊆ B. True or false? Give a counterexample if false.

Explore the statement 'If n(A) < n(B), then A ⊆ B.' and discover whether it's true or false with a detailed explanation and counterexample. Understand key concepts in set theory, including cardinality and subsets. Suitable for high school students in Grades 9-12.

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