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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) = 17,n(B)=23 and n(A cup B) = 38`, then find `n(A cap 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. The formula is: \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] 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:** - \( n(A) = 17 \) - \( n(B) = 23 \) - \( n(A \cup B) = 38 \) 2. **Substitute the values into the formula:** \[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \] Plugging in the known values: \[ 38 = 17 + 23 - n(A \cap B) \] 3. **Calculate the sum of \( n(A) \) and \( n(B) \):** \[ 17 + 23 = 40 \] So, we have: \[ 38 = 40 - n(A \cap B) \] 4. **Rearrange the equation to solve for \( n(A \cap B) \):** \[ n(A \cap B) = 40 - 38 \] 5. **Perform the subtraction:** \[ n(A \cap B) = 2 \] ### Final Answer: The number of elements in the intersection of sets A and B is \( n(A \cap B) = 2 \). ---
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