If `A={2,3,4,5,6}`, Then which of the following is not true ?

Options

A. `EEx""inA,` such that `x+3=8`
B. `EEx""inA,` such that `x+2lt5`
C. `AAx""inA`, such that `x+2lt9`
D. `AAx""inA,` such that `x+6ge9`

Answer

To determine which statement is not true for the set \( A = \{2, 3, 4, 5, 6\} \), we will analyze each option step by step.

### Step 1: Analyze the first option
**Statement:** For exactly one \( x \) such that \( x \in A \), \( x + 3 = 8 \).

- Solve for \( x \):
\[ x + 3 = 8 \implies x = 8 - 3 = 5 \]
- Check if \( 5 \) is in the set \( A \):
- Yes, \( 5 \in A \).
- Conclusion: There is exactly one \( x \) (which is \( 5 \)) that satisfies this condition. **This statement is true.**

### Step 2: Analyze the second option
**Statement:** For exactly one \( x \) such that \( x \in A \), \( x + 2 < 5 \).

- Solve for \( x \):
\[ x + 2 < 5 \implies x < 5 - 2 \implies x < 3 \]
- Check the elements of \( A \):
- The only element less than \( 3 \) is \( 2 \):
- \( 2 + 2 = 4 < 5 \) (True)
- Check other elements:
- \( 3 + 2 = 5 \) (Not less than \( 5 \))
- \( 4 + 2 = 6 \) (Not less than \( 5 \))
- \( 5 + 2 = 7 \) (Not less than \( 5 \))
- \( 6 + 2 = 8 \) (Not less than \( 5 \))
- Conclusion: There is exactly one \( x \) (which is \( 2 \)) that satisfies this condition. **This statement is true.**

### Step 3: Analyze the third option
**Statement:** For all \( x \in A \), \( x + 2 < 9 \).

- Check each element of \( A \):
- For \( x = 2 \): \( 2 + 2 = 4 < 9 \) (True)
- For \( x = 3 \): \( 3 + 2 = 5 < 9 \) (True)
- For \( x = 4 \): \( 4 + 2 = 6 < 9 \) (True)
- For \( x = 5 \): \( 5 + 2 = 7 < 9 \) (True)
- For \( x = 6 \): \( 6 + 2 = 8 < 9 \) (True)
- Conclusion: All elements satisfy the condition. **This statement is true.**

### Step 4: Analyze the fourth option
**Statement:** For all \( x \in A \), \( x + 6 \geq 9 \).

- Check each element of \( A \):
- For \( x = 2 \): \( 2 + 6 = 8 \geq 9 \) (False)
- For \( x = 3 \): \( 3 + 6 = 9 \geq 9 \) (True)
- For \( x = 4 \): \( 4 + 6 = 10 \geq 9 \) (True)
- For \( x = 5 \): \( 5 + 6 = 11 \geq 9 \) (True)
- For \( x = 6 \): \( 6 + 6 = 12 \geq 9 \) (True)
- Conclusion: The statement fails for \( x = 2 \). **This statement is false.**

### Final Conclusion
The statement that is **not true** is the fourth option: "For all \( x \in A \), \( x + 6 \geq 9 \)".