If n (X) = 10 , n(Y) = 5 , and the sets X and Y are dijoint , then find `n(X cup Y)` .

Answer

To solve the problem, we need to find the number of elements in the union of two disjoint sets, X and Y.

### Step-by-Step Solution:

1. **Identify the number of elements in each set:**
- We are given that \( n(X) = 10 \) (the number of elements in set X).
- We are also given that \( n(Y) = 5 \) (the number of elements in set Y).

2. **Understand the concept of disjoint sets:**
- Disjoint sets are sets that have no elements in common. This means that the intersection of the two sets is empty.
- Therefore, \( n(X \cap Y) = 0 \).

3. **Use the formula for the union of two sets:**
- The formula for the number of elements in the union of two sets is given by:
\[ n(X \cup Y) = n(X) + n(Y) - n(X \cap Y) \]
- Since \( n(X \cap Y) = 0 \) for disjoint sets, the formula simplifies to:
\[ n(X \cup Y) = n(X) + n(Y) \]

4. **Substitute the values into the formula:**
- Now, substitute the known values into the simplified formula:
\[ n(X \cup Y) = 10 + 5 \]

5. **Calculate the result:**
- Performing the addition gives:
\[ n(X \cup Y) = 15 \]

### Final Answer:
The number of elements in the union of sets X and Y is \( n(X \cup Y) = 15 \).

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