Let A and B be two finite sets such that `n(A) = 20, n(B) = 28` and `n(A ∪ B)=36`, then find `n(A ∩ B)`.

Answer

To solve the problem, we will 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:
- \( n(A) \) is the number of elements in set A,
- \( n(B) \) is the number of elements in set B,
- \( n(A \cup B) \) is the number of elements in the union of sets A and B,
- \( n(A \cap B) \) is the number of elements in the intersection of sets A and B.

Given:
- \( n(A) = 20 \)
- \( n(B) = 28 \)
- \( n(A \cup B) = 36 \)

We need to find \( n(A \cap B) \).

### Step 1: Write down the formula
We start with the formula for the union of two sets:
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

### Step 2: Substitute the known values
Substituting the known values into the formula:
\[ 36 = 20 + 28 - n(A \cap B) \]

### Step 3: Simplify the equation
Now, simplify the right side:
\[ 36 = 48 - n(A \cap B) \]

### Step 4: Rearrange the equation
Rearranging the equation to isolate \( n(A \cap B) \):
\[ n(A \cap B) = 48 - 36 \]

### Step 5: Calculate the value
Now, perform the calculation:
\[ n(A \cap B) = 12 \]

Thus, the number of elements in the intersection of sets A and B is \( n(A \cap B) = 12 \).

### Final Answer:
The value of \( n(A \cap B) \) is **12**.

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