If n(A) = 110, n(B) = 105, and n(A ∪ B) = 195, what is n(A ∩ B)?

To find \( n(A \cap B) \), we use the formula for the union of two sets:

\[
n(A \cup B) = n(A) + n(B) - n(A \cap B)
\]

We are given the following values:
- \( n(A) = 110 \)
- \( n(B) = 105 \)
- \( n(A \cup B) = 195 \)

Now, substitute these values into the formula:

\[
195 = 110 + 105 - n(A \cap B)
\]

Simplify the equation:

\[
195 = 215 - n(A \cap B)
\]

Solve for \( n(A \cap B) \):

\[
n(A \cap B) = 215 - 195 = 20
\]

Thus, \( n(A \cap B) = 20 \).

Would you like further explanation or have any other questions?

Here are 5 related questions:
1. If \( n(A \cup B) = 250 \), \( n(A) = 150 \), and \( n(A \cap B) = 50 \), what is \( n(B) \)?
2. How would you find \( n(A \cup B \cup C) \) if you had three sets involved?
3. If \( n(A \cup B) = n(A) + n(B) \), what can you infer about \( A \) and \( B \)?
4. If \( A \) and \( B \) are mutually exclusive, what is \( n(A \cap B) \)?
5. How does the inclusion-exclusion principle apply when dealing with more than two sets?

**Tip**: Always visualize sets with Venn diagrams when working with set operations for a clearer understanding of their relationships.

Discover how to find the intersection of two sets using set theory. Given n(A) = 110, n(B) = 105, and n(A ∪ B) = 195, this step-by-step solution explains how to calculate n(A ∩ B) using the inclusion-exclusion principle. Ideal for students in Grades 9-11.

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