In sub - part (i) and (ii) choose the correct option and in sub - part (iii) to (v) answer the questions as instructed.
`~qto~p` is equivalent to ……

Answer

To solve the problem of finding the equivalence of `~q to ~p`, we will use a truth table method. Here’s a step-by-step solution:

### Step 1: Define the Variables
Let \( p \) and \( q \) be two propositions. We will evaluate the negations and the conditional relationship between them.

### Step 2: Create a Truth Table
We will create a truth table for the propositions \( p \) and \( q \), their negations \( \neg p \) and \( \neg q \), and the conditional \( \neg q \to \neg p \).

| \( p \) | \( q \) | \( \neg p \) | \( \neg q \) | \( \neg q \to \neg p \) |
|---------|---------|---------------|---------------|--------------------------|
| T | T | F | F | T |
| T | F | F | T | F |
| F | T | T | F | T |
| F | F | T | T | T |

### Step 3: Analyze the Truth Table
From the truth table, we can analyze the column for \( \neg q \to \neg p \):
- The only time \( \neg q \to \neg p \) is false is when \( \neg q \) is true and \( \neg p \) is false, which occurs when \( q \) is false and \( p \) is true.

### Step 4: Conclusion
The conditional \( \neg q \to \neg p \) is equivalent to the conditional \( p \to q \). Therefore, we can conclude that:

\[ \neg q \to \neg p \text{ is equivalent to } p \to q \]

### Final Answer
The expression `~q to ~p` is equivalent to `p to q`.

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