Two sets A and B are such that `n(A cup B)=21 ,n(A)=10,n(B)=15` then find `n(a cap B) and n(A-B)`

Answer

To solve the problem, we need to find the number of elements in the intersection of sets A and B (denoted as \( n(A \cap B) \)) and the number of elements in the difference of set A and set B (denoted as \( n(A - B) \)).

Given:
- \( n(A \cup B) = 21 \)
- \( n(A) = 10 \)
- \( n(B) = 15 \)

### Step 1: Use the formula for the union of two sets
The formula for the union of two sets is given by:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
Substituting the known values into the formula:
\[ 21 = 10 + 15 - n(A \cap B) \]

### Step 2: Simplify the equation
Now, simplify the equation:
\[ 21 = 25 - n(A \cap B) \]

### Step 3: Solve for \( n(A \cap B) \)
Rearranging the equation to find \( n(A \cap B) \):
\[ n(A \cap B) = 25 - 21 \]
\[ n(A \cap B) = 4 \]

### Step 4: Find \( n(A - B) \)
To find \( n(A - B) \), we use the formula:
\[ n(A - B) = n(A) - n(A \cap B) \]
Substituting the known values:
\[ n(A - B) = 10 - 4 \]
\[ n(A - B) = 6 \]

### Final Answers:
- \( n(A \cap B) = 4 \)
- \( n(A - B) = 6 \)