If X and Y are two sets such that `XuuY` has 17 elements, X has 8 elements and Y has 14 elements . Then the number of elements of `XcapY`= …………..

Answer

To find the number of elements in the intersection of sets \(X\) and \(Y\), we can use the principle of inclusion-exclusion for sets. The formula we will use is:

\[ |X \cup Y| = |X| + |Y| - |X \cap Y| \]

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

### Step-by-step Solution:

1. **Identify the given values:**
- \(|X \cup Y| = 17\)
- \(|X| = 8\)
- \(|Y| = 14\)

2. **Substitute the values into the formula:**
\[ 17 = 8 + 14 - |X \cap Y| \]

3. **Simplify the equation:**
- First, calculate \(8 + 14\):
\[ 8 + 14 = 22 \]
- Now substitute back into the equation:
\[ 17 = 22 - |X \cap Y| \]

4. **Rearrange the equation to solve for \(|X \cap Y|\):**
\[ |X \cap Y| = 22 - 17 \]

5. **Calculate the value:**
\[ |X \cap Y| = 5 \]

### Final Answer:
The number of elements in the intersection of sets \(X\) and \(Y\) is \(5\).

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