p : if a natural is odd , then its sqare also odd. Which of the following does not convey the same meaning as that by statement p

Options

A. A natural number is odd implies that its square is also odd
B. For a natural number to be odd it is necessary that its square is odd
C. If the square of a natural number is not even then it is odd
D. If the square of natural number is not odd then it is odd

Answer

To solve the problem, we need to analyze the statement \( p \) and determine which of the given options does not convey the same meaning. The statement \( p \) is: "If a natural number is odd, then its square is also odd."

### Step 1: Understand the Original Statement
The original statement can be expressed in logical terms as:
- Let \( x \): "A natural number is odd."
- Let \( y \): "The square of the natural number is odd."

Thus, the statement \( p \) can be written as:
\[ x \implies y \]
This means if \( x \) is true (the number is odd), then \( y \) must also be true (its square is odd).

### Step 2: Analyze the Options
We need to evaluate each option to see if it conveys the same meaning as \( p \).

1. **Option 1**: "A natural number is odd implies that its square is also odd."
- This is a direct restatement of \( p \) and thus conveys the same meaning. **(True)**

2. **Option 2**: "For a natural number to be odd, it is necessary that its square is odd."
- This can be interpreted as saying that if a number is odd, then its square must be odd, which is equivalent to \( p \). **(True)**

3. **Option 3**: "If the square of a natural number is even, then the number is not odd."
- This statement is logically equivalent to saying that if the square is not odd (i.e., even), then the number must be even (not odd). This does not contradict \( p \) and is true in the context of natural numbers. **(True)**

4. **Option 4**: "If the square of a natural number is not odd, then it is odd."
- This statement is logically incorrect. If the square is not odd (meaning it is even), it cannot be odd at the same time. Therefore, this option does not convey the same meaning as \( p \). **(False)**

### Conclusion
The option that does not convey the same meaning as the statement \( p \) is **Option 4**.