If a set A contain 9 elements and set B contains 5 elements, then which of the following is not true?

Options

A. `n(A cup B) ge 9`
B. `n(A cup B) le 14`
C. `n(a cap B) le 5`
D. `n(A cap B) ge 9`

Answer

To solve the problem, we need to analyze the properties of the sets A and B given their sizes.

1. **Identify the sizes of the sets**:
- Set A has 9 elements: \( n(A) = 9 \)
- Set B has 5 elements: \( n(B) = 5 \)

2. **Determine the possible values for the union of sets A and B**:
- The maximum number of elements in the union of two sets occurs when there are no common elements. Thus, the maximum value of \( n(A \cup B) \) is:
\[ n(A \cup B) = n(A) + n(B) = 9 + 5 = 14 \]
- The minimum number of elements in the union occurs when all elements of set B are also in set A. Thus, the minimum value of \( n(A \cup B) \) is:
\[ n(A \cup B) = n(A) = 9 \]
- Therefore, the range of \( n(A \cup B) \) is:
\[ 9 \leq n(A \cup B) \leq 14 \]

3. **Determine the possible values for the intersection of sets A and B**:
- The maximum number of elements in the intersection occurs when all elements of set B are also in set A. Thus, the maximum value of \( n(A \cap B) \) is:
\[ n(A \cap B) = n(B) = 5 \]
- The minimum number of elements in the intersection occurs when there are no common elements. Thus, the minimum value of \( n(A \cap B) \) is:
\[ n(A \cap B) = 0 \]
- Therefore, the range of \( n(A \cap B) \) is:
\[ 0 \leq n(A \cap B) \leq 5 \]

4. **Evaluate the options**:
- **Option A**: \( n(A \cup B) \geq 9 \) - This is true since the minimum is 9.
- **Option B**: \( n(A \cup B) < 14 \) - This is also true since the maximum is 14.
- **Option C**: \( n(A \cap B) < 5 \) - This is true because the maximum is 5.
- **Option D**: \( n(A \cap B) \geq 9 \) - This is false because the maximum intersection can only be 5.

5. **Conclusion**:
- The statement that is not true is **Option D**: \( n(A \cap B) \geq 9 \).