If X and Y are two sets such that `(XuuY)` has 60 elements, X has 38 elements and Y has 42 elements, how many elements does `(XnnY)` have?

Options

A. 11
B. 20
C. 13
D. None of these

Answer

To find the number of elements in the intersection of sets \(X\) and \(Y\), we can use the formula for the union of two sets:

\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]

Where:
- \(n(X \cup Y)\) is the number of elements in the union of sets \(X\) and \(Y\).
- \(n(X)\) is the number of elements in set \(X\).
- \(n(Y)\) is the number of elements in set \(Y\).
- \(n(X \cap Y)\) is the number of elements in the intersection of sets \(X\) and \(Y\).

Given:
- \(n(X \cup Y) = 60\)
- \(n(X) = 38\)
- \(n(Y) = 42\)

We need to find \(n(X \cap Y)\).

### Step 1: Substitute the known values into the formula

\[ 60 = 38 + 42 - n(X \cap Y) \]

### Step 2: Simplify the equation

Combine the numbers on the right side:

\[ 60 = 80 - n(X \cap Y) \]

### Step 3: Rearrange the equation to solve for \(n(X \cap Y)\)

\[ n(X \cap Y) = 80 - 60 \]

### Step 4: Calculate the value

\[ n(X \cap Y) = 20 \]

Thus, the number of elements in the intersection of sets \(X\) and \(Y\) is \(20\).

### Final Answer:
The number of elements in \(X \cap Y\) is \(20\).

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