If A and B are two sets such that A has 12 elements, B has 17 elements, and `A uu B` has 21 elements, how many elements docs `A nn B` have?

Options

A. 7
B. 8
C. 9
D. 10

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:

\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]

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 1: Identify the given values
From the problem, we have:
- \(N(A) = 12\) (the number of elements in set A)
- \(N(B) = 17\) (the number of elements in set B)
- \(N(A \cup B) = 21\) (the number of elements in the union of sets A and B)

### Step 2: Substitute the values into the formula
Now, we substitute the known values into the formula:

\[ 21 = 12 + 17 - N(A \cap B) \]

### Step 3: Simplify the equation
Combine the numbers on the right side:

\[ 21 = 29 - N(A \cap B) \]

### Step 4: Solve for \(N(A \cap B)\)
To find \(N(A \cap B)\), we rearrange the equation:

\[ N(A \cap B) = 29 - 21 \]

Calculating the right side gives:

\[ N(A \cap B) = 8 \]

### Conclusion
Thus, the number of elements in the intersection of sets A and B is \(8\).