Negation of the statement if a number is prime then it is odd' is.

Options

A. A number is not prime but odd.
B. A number prime and it is not odd.
C. A number is naither primes nor odd
D. None of these

Answer

To find the negation of the statement "If a number is prime, then it is odd," we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Components of the Statement**:
- Let \( p \) be the statement "a number is prime."
- Let \( q \) be the statement "it is odd."
- The original statement can be expressed as \( p \implies q \) (if \( p \), then \( q \)).

2. **Understand the Negation of Implication**:
- The negation of the implication \( p \implies q \) is given by \( \neg(p \implies q) \).
- According to logical equivalences, \( p \implies q \) is equivalent to \( \neg p \lor q \) (not \( p \) or \( q \)).
- Therefore, the negation \( \neg(p \implies q) \) can be rewritten as \( \neg(\neg p \lor q) \).

3. **Apply De Morgan's Laws**:
- Using De Morgan's Laws, we can simplify \( \neg(\neg p \lor q) \) to \( p \land \neg q \) (which means \( p \) is true and \( q \) is false).

4. **Translate Back to Verbal Form**:
- The expression \( p \land \neg q \) translates back to: "A number is prime and it is not odd."

5. **Final Statement**:
- Therefore, the negation of the statement "If a number is prime, then it is odd" is: "A number is prime and it is not odd."