If n (A) = 4 , n (B) = 6 , and n `( A cup B) =8 ` , then find `n(A cap B)` .

Answer

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.

Given:
- \( n(A) = 4 \)
- \( n(B) = 6 \)
- \( n(A \cup B) = 8 \)

Now, we can substitute the values into the formula:

1. Substitute the values into the formula:
\[ 8 = 4 + 6 - n(A \cap B) \]

2. Simplify the equation:
\[ 8 = 10 - n(A \cap B) \]

3. Rearrange the equation to solve for \( n(A \cap B) \):
\[ n(A \cap B) = 10 - 8 \]

4. Calculate the value:
\[ n(A \cap B) = 2 \]

Thus, the number of elements in the intersection of sets A and B is \( n(A \cap B) = 2 \).