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

Options

A. 25
B. 35
C. 30
D. 40

Answer

To find the number of elements in set 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:
- \( n(X) = 40 \)
- \( n(X \cup Y) = 60 \)
- \( n(X \cap Y) = 10 \)

We can substitute these values into the formula:

1. Substitute the known values into the formula:
\[ 60 = 40 + n(Y) - 10 \]

2. Simplify the equation:
\[ 60 = 30 + n(Y) \]

3. Solve for \( n(Y) \):
\[ n(Y) = 60 - 30 \]
\[ n(Y) = 30 \]

Thus, the number of elements in set Y is **30**.