Let `p, q and r` be three statements, then `(~p to q) to r` is equivalent to

Options

A. `(p vv r) ^^ (q vv r)`
B. `(~p vv r) ^^ (~q vv r)`
C. `(p ^^ r) vv (q vv r)`
D. `(p vv r) to r`

Answer

To solve the problem, we need to determine the equivalence of the expression \( (~p \to q) \to r \). We will use logical equivalences and transformations step by step.

### Step-by-Step Solution:

1. **Understanding the Implication**:
The expression \( A \to B \) can be rewritten using logical equivalences as \( \neg A \lor B \). Therefore, we can rewrite \( (~p \to q) \) as:
\[ \neg (~p) \lor q \]
This simplifies to:
\[ p \lor q \]

2. **Rewriting the Entire Expression**:
Now, we substitute this back into the original expression:
\[ (p \lor q) \to r \]
Again, using the implication equivalence, we rewrite this as:
\[ \neg (p \lor q) \lor r \]

3. **Applying De Morgan's Law**:
We can apply De Morgan's Law to simplify \( \neg (p \lor q) \):
\[ \neg (p \lor q) = \neg p \land \neg q \]
Therefore, we can rewrite the expression as:
\[ (\neg p \land \neg q) \lor r \]

4. **Final Form**:
The expression \( (\neg p \land \neg q) \lor r \) is the final simplified form of the original expression \( (~p \to q) \to r \).

### Conclusion:
Thus, the expression \( (~p \to q) \to r \) is equivalent to:
\[ (\neg p \land \neg q) \lor r \]