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If A and B are two sets such that n(A) =...

If A and B are two sets such that `n(A) =8, n(B)=11 and n(A cup B) =14` then find ` n(Acap B)`.

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To find the number of elements in the intersection of sets A and B, we can use the formula for the union of two sets: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Where: - \( n(A \cup B) \) is the number of elements in the union of sets A and B. - \( n(A) \) is the number of elements in set A. - \( n(B) \) is the number of elements in set B. - \( n(A \cap B) \) is the number of elements in the intersection of sets A and B. ### Step-by-step Solution: 1. **Identify the given values:** - \( n(A) = 8 \) - \( n(B) = 11 \) - \( n(A \cup B) = 14 \) 2. **Substitute the known values into the union formula:** \[ 14 = 8 + 11 - n(A \cap B) \] 3. **Simplify the equation:** \[ 14 = 19 - n(A \cap B) \] 4. **Rearrange to isolate \( n(A \cap B) \):** \[ n(A \cap B) = 19 - 14 \] 5. **Calculate \( n(A \cap B) \):** \[ n(A \cap B) = 5 \] ### Final Answer: The number of elements in the intersection of sets A and B is \( n(A \cap B) = 5 \). ---
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