If X and Y are two sets such that X has 40 elements, `X uuY`has 60 elements and `X nnY`has 10 elements, how many elements does Y have?

Answer

To solve the problem, we will use the formula for the number of elements in the union of two sets. The formula is given by:

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

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:
- \( n(X) = 40 \)
- \( n(X \cup Y) = 60 \)
- \( n(X \cap Y) = 10 \)

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

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

\[ 60 = 40 + n(Y) - 10 \]

### Step 2: Simplify the equation.

Combine the constants on the right side:

\[ 60 = 30 + n(Y) \]

### Step 3: Solve for \( n(Y) \).

Subtract 30 from both sides:

\[ n(Y) = 60 - 30 \]

\[ n(Y) = 30 \]

### Conclusion:

The number of elements in set Y is \( 30 \).

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