Consider the proposition : " if the pressure increases, the volume decreases". The negation of this propositions is

Options

A. If the pressure does not increase the volume does not decrease
B. of the volume increases, the pressure decreases,
C. if the volume does not decrease, the pressure, does not increase
D. If the volume decreases, then the pressure increases.

Answer

To find the negation of the proposition "if the pressure increases, the volume decreases," we can follow these steps:

### Step 1: Identify the Statements
Let:
- \( P \): "The pressure increases."
- \( Q \): "The volume decreases."

### Step 2: Write the Proposition in Symbolic Form
The given proposition can be expressed in symbolic form as:
\[ P \implies Q \]
This reads as "If \( P \) (the pressure increases), then \( Q \) (the volume decreases)."

### Step 3: Find the Negation of the Implication
The negation of the implication \( P \implies Q \) can be expressed using the logical equivalence:
\[ \neg (P \implies Q) \equiv P \land \neg Q \]
This means that the negation of "if \( P \) then \( Q \)" is equivalent to \( P \) being true and \( Q \) being false.

### Step 4: Write the Negation in Statement Form
Now we need to express \( P \land \neg Q \) in words:
- \( P \): "The pressure increases."
- \( \neg Q \): "The volume does not decrease."

Thus, the negation of the original proposition can be stated as:
"The pressure increases and the volume does not decrease."

### Final Answer
The negation of the proposition "if the pressure increases, the volume decreases" is:
"The pressure increases and the volume does not decrease."

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