The negation of the proposition " If 2 is prime then 3 is odd" is

Options

A. If 2 is not prime then 3 is not odd
B. 2 is prime and 3 is not odd
C. 2 is not prime and 3 is odd
D. If 2 not prime then 3 is odd

Answer

To find the negation of the proposition "If 2 is prime then 3 is odd," we can follow these steps:

### Step 1: Identify the components of the proposition
Let:
- \( P \): "2 is prime"
- \( Q \): "3 is odd"

The original proposition can be expressed in logical terms as:
\[ P \implies Q \]
This reads as "If \( P \) then \( Q \)."

### Step 2: Understand 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 " \( P \) is true and \( Q \) is false."

### Step 3: Apply the components to the negation
Now, substituting back the definitions of \( P \) and \( Q \):
- \( P \): "2 is prime" (which is true)
- \( \neg Q \): "3 is not odd" (which means 3 is even)

Thus, the negation of the original proposition becomes:
\[ P \land \neg Q \]
This translates to:
"2 is prime and 3 is not odd."

### Step 4: Write the final answer
The final answer is:
"2 is prime and 3 is not odd."