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!