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Let n(U) = 700, n(A) = 200, n(B) = 300 a...

Let `n(U) = 700, n(A) = 200, n(B) = 300 and n(AuuB) = 400.` Then `n(A^c uu B^c)` is equal to

A

100

B

500

C

300

D

600

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find \( n(A^c \cup B^c) \). We can use the principle of set theory and the relationships between the sets. ### Step-by-Step Solution: 1. **Understanding the given values**: - \( n(U) = 700 \) (Total number of elements in the universal set) - \( n(A) = 200 \) (Number of elements in set A) - \( n(B) = 300 \) (Number of elements in set B) - \( n(A \cup B) = 400 \) (Number of elements in the union of sets A and B) 2. **Using the formula for the union of two sets**: The formula for the union of two sets is: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Plugging in the values we have: \[ 400 = 200 + 300 - n(A \cap B) \] 3. **Solving for \( n(A \cap B) \)**: Rearranging the equation to find \( n(A \cap B) \): \[ n(A \cap B) = 200 + 300 - 400 = 100 \] 4. **Finding \( n(A^c \cup B^c) \)**: We can use De Morgan's law, which states: \[ A^c \cup B^c = (A \cap B)^c \] Therefore, we can find \( n(A^c \cup B^c) \) using: \[ n(A^c \cup B^c) = n(U) - n(A \cap B) \] Substituting the values we have: \[ n(A^c \cup B^c) = 700 - 100 = 600 \] 5. **Conclusion**: Thus, the value of \( n(A^c \cup B^c) \) is \( 600 \). ### Final Answer: \[ n(A^c \cup B^c) = 600 \]
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