Let S be non-empty subset of R. consider the following statement:
P: There is a rational number ` x ne S " such that " x gt 0`
Which of the following statements is the negation of the statement P ?

Options

A. Every rational number `x ne S ` such that `x le 0`
B. `x in S and x le 0 Rightarrow x` is not rational
C. There is a rational number `x in S " such that " x le 0`
D. There is no rational number `x in S " such that x le 0`

Explanation

To find the negation of the statement P: "There is a rational number \( x \notin S \) such that \( x > 0 \)", we will follow these steps:

### Step 1: Identify the structure of the statement
The original statement P can be broken down as:
- There exists a rational number \( x \) (denoted as \( \exists x \in \mathbb{Q} \))
- Such that \( x \notin S \)
- And \( x > 0 \)

### Step 2: Apply the negation
To negate the statement, we need to change the existential quantifier \( \exists \) to a universal quantifier \( \forall \) and negate the conditions that follow.

The negation of the statement can be expressed as:
- For every rational number \( x \) (denoted as \( \forall x \in \mathbb{Q} \))
- It is not true that \( x \notin S \) (which means \( x \in S \))
- Or \( x \leq 0 \) (negating \( x > 0 \))

### Step 3: Combine the negated conditions
Putting it all together, the negation of statement P is:
- "For every rational number \( x \), if \( x \notin S \), then \( x \leq 0 \)."

### Step 4: Identify the correct option
From the options given, we need to find the one that matches our negated statement.

The correct negation of statement P is:
- "For every rational number \( x \), \( x \in S \) or \( x \leq 0 \)."

### Final Answer
The correct option that represents the negation of statement P is:
- **Option C**: "There is a rational number \( x \in S \) such that \( x \leq 0 \)."