If n(U) = 200 and n(A∪B) = 150, what is n(Ac ∩
Bc)?

Given:

- \( n(U) = 200 \) (the total number of elements in the universal set \( U \)),
- \( n(A \cup B) = 150 \) (the number of elements in the union of sets \( A \) and \( B \)).

We are asked to find \( n(A^c \cap B^c) \), which represents the number of elements in the intersection of the complements of sets \( A \) and \( B \). This intersection is also the complement of \( A \cup B \), denoted as \( (A \cup B)^c \).

Using the principle of complements:

\[
n(A^c \cap B^c) = n((A \cup B)^c) = n(U) - n(A \cup B)
\]

Substituting the given values:

\[
n(A^c \cap B^c) = 200 - 150 = 50
\]

So, \( n(A^c \cap B^c) = 50 \).

Would you like more details or have any questions?

Here are 5 related questions:

1. What is the principle of inclusion-exclusion?
2. How do you calculate \( n(A \cap B) \) given \( n(A) \), \( n(B) \), and \( n(A \cup B) \)?
3. What is the significance of complements in set theory?
4. Can you explain the difference between \( A \cup B \) and \( A \cap B \)?
5. How do Venn diagrams help in understanding set operations?

**Tip:** When working with complements, remember that the complement of a union is the intersection of the complements, and vice versa.

Explore how to find n(Ac ∩ Bc) in set theory with a step-by-step solution. Given n(U) = 200 and n(A∪B) = 150, learn how to apply the principle of complements to solve for n(Ac ∩ Bc). This problem involves understanding complements, intersections, and unions in set theory.

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