We are given:

• $n(A) = 15$
• $n(B) = 30$
• $n(A \cup B) = 46$

We are asked to find $n(A \cap B)$, the number of elements in the intersection of sets $A$ and $B$.

Using the principle of inclusion-exclusion, we know that:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Substituting the given values into this formula:

$46 = 15 + 30 - n(A \cap B)$

Simplifying:

$46 = 45 - n(A \cap B)$

Now, solve for $n(A \cap B)$:

$n(A \cap B) = 45 - 46 = -1$

However, a negative value is not possible for the cardinality of a set. This indicates that there's an inconsistency in the given information. Specifically, the condition $n(A \cup B) = 46$ is problematic because the union of two sets cannot have more elements than the total combined elements of both sets. In this case, the maximum possible value for $n(A \cup B)$ should be $n(A) + n(B) = 15 + 30 = 45$.

Would you like me to adjust the numbers or clarify this? Let me know if you'd like more details or further questions.

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Here are 5 related questions:

1. What if $n(A \cup B)$ was 45 instead of 46?
2. How does the principle of inclusion-exclusion apply in set theory?
3. Can $n(A \cap B)$ ever be negative?
4. How do you calculate the size of the intersection of two sets from their union?
5. What changes if sets $A$ and $B$ are disjoint?

Tip: