Which of the following is the negation of the statement "for all ` M gt 0` , there exists ` x in S ` such that ` x ge M"` ?

Options

A. there exists ` M gt 0`, such that ` x lt ` M for all ` x in S `
B. there exists ` M gt 0`, there exists ` x in S ` such that ` x ge M`
C. there exists `M gt 0`, there exists ` x in S ` such that ` x lt M`
D. there exists ` M gt 0` , such that ` x ge M` for all ` x in S `

Answer

To find the negation of the statement "for all \( M > 0 \), there exists \( x \in S \) such that \( x \geq M \)", we can follow these steps:

### Step 1: Understand the Original Statement
The original statement can be broken down as:
- "For all \( M > 0 \)" means that the statement applies to every positive value of \( M \).
- "There exists \( x \in S \)" means that for each \( M \), we can find at least one \( x \) in the set \( S \).
- "Such that \( x \geq M \)" means that this \( x \) is greater than or equal to \( M \).

### Step 2: Identify the Structure of the Statement
The statement has the form:
- Universal quantifier (for all) followed by an existential quantifier (there exists).

### Step 3: Apply Negation Rules
The negation of a statement that has the structure "for all \( M \), there exists \( x \)" can be transformed using the following rules:
- The negation of "for all" (\( \forall \)) is "there exists" (\( \exists \)).
- The negation of "there exists" (\( \exists \)) is "for all" (\( \forall \)).
- Additionally, we need to negate the inner condition.

### Step 4: Write the Negation
Applying the negation:
1. Start with "for all \( M > 0 \)" and negate it to "there exists \( M > 0 \)".
2. Then negate "there exists \( x \in S \) such that \( x \geq M \)" to "for all \( x \in S \), \( x < M \)".

Combining these, we get:
"There exists \( M > 0 \) such that for all \( x \in S \), \( x < M \)".

### Step 5: Final Statement
Thus, the negation of the original statement is:
"There exists \( M > 0 \) such that for all \( x \in S \), \( x < M \)".