Two sets A and B are such that `n(A cup B)=21,n(A' cap B')=9,n(A cap B)=7` find `n(A cap B)'`

Answer

To solve the problem step by step, we will use the information provided about the sets A and B.

### Given:
1. \( n(A \cup B) = 21 \)
2. \( n(A' \cap B') = 9 \) (This represents the number of elements not in A and not in B)
3. \( n(A \cap B) = 7 \)

We need to find \( n(A \cap B)' \) (the number of elements not in \( A \cap B \)).

### Step 1: Find the number of elements in the universal set \( n(U) \)

Using the relationship between the complements, we know that:
\[ n(A' \cap B') = n(U) - n(A \cup B) \]
Substituting the known values:
\[ 9 = n(U) - 21 \]
To find \( n(U) \), we rearrange the equation:
\[ n(U) = 9 + 21 = 30 \]

### Step 2: Find \( n(A \cap B)' \)

Using the formula for the complement:
\[ n(A \cap B)' = n(U) - n(A \cap B) \]
Substituting the values we have:
\[ n(A \cap B)' = 30 - 7 = 23 \]

### Conclusion

Thus, the number of elements not in \( A \cap B \) is:
\[ n(A \cap B)' = 23 \]