If `x and y` are positive real number, then :

Options

A. `x gt y implies -x gt -y`
B. `x gt y implies -x lt - y`
C. `x gt y implies 1/x gt 1/y`
D. `x gt y implies (-1)/x lt (-1)/y`

Answer

To solve the problem, we need to analyze the given statements regarding the positive real numbers \( x \) and \( y \).

### Step-by-Step Solution:

1. **Understanding the Definitions**:
- Positive real numbers are any numbers greater than zero. This means \( x > 0 \) and \( y > 0 \).

2. **Analyzing the Statements**:
- We are given a few statements (options) to evaluate whether they are true or false based on the conditions that \( x \) and \( y \) are positive real numbers.

3. **Evaluating Option 1**:
- The statement is: If \( x > y \), then \( -x > -y \).
- Let's assume \( x = 2 \) and \( y = 1 \) (since \( x > y \)).
- Therefore, \( -x = -2 \) and \( -y = -1 \).
- Now, we check: Is \( -2 > -1 \)?
- This is false since \(-2\) is less than \(-1\).
- Hence, **Option 1 is incorrect**.

4. **Evaluating Option 2**:
- The statement is: If \( x > y \), then \( -x < -y \).
- Using the same values \( x = 2 \) and \( y = 1 \):
- We have \( -x = -2 \) and \( -y = -1 \).
- Now, we check: Is \( -2 < -1 \)?
- This is true since \(-2\) is indeed less than \(-1\).
- Hence, **Option 2 is correct**.

5. **Conclusion**:
- Since we found that Option 2 is correct, we can conclude that it is the valid statement given the conditions.