Which of the following is true?
(i) If p is a statement then ~p is not a statement
(ii) If p is a statement then ~p is also a statement
(iii) Negation of p : 0 is a positive number is, 0 is a negative number

Options

A. Only (ii)
B. Only (i)
C. (i) & (iii)
D. (ii) & (iii)

Answer

To determine which of the given statements is true, let's analyze each statement one by one.

**Step 1: Analyze Statement (i)**
Statement (i) says: "If p is a statement then ~p is not a statement."
- Let's take an example where p is "Today is Sunday." This is a statement.
- The negation of p, denoted as ~p, would be "Today is not Sunday."
- Since "Today is not Sunday" is also a statement, this means that ~p is indeed a statement.
- Therefore, statement (i) is **false**.

**Step 2: Analyze Statement (ii)**
Statement (ii) says: "If p is a statement then ~p is also a statement."
- Using the same example, if p is "Today is Sunday," then ~p is "Today is not Sunday."
- As established, both p and ~p are statements.
- Thus, statement (ii) is **true**.

**Step 3: Analyze Statement (iii)**
Statement (iii) says: "Negation of p: 0 is a positive number is, 0 is a negative number."
- The statement "0 is a positive number" is indeed false.
- The negation of this statement would be "0 is not a positive number," which is true since 0 is neither positive nor negative.
- The statement "0 is a negative number" is also false.
- Therefore, statement (iii) is **false**.

**Conclusion:**
The only true statement among the three is statement (ii): "If p is a statement then ~p is also a statement."