The logical statement `(~ q implies ~p ) vv (~ q implies p)` is equivalent to :

Options

A. `p to q`
B. `q to p`
C. `~ (~p to q)`
D. Tautology

Explanation

To solve the logical statement \( (~q \implies \sim p) \lor (~q \implies p) \), we can follow these steps:

### Step 1: Rewrite Implications
The implication \( A \implies B \) can be rewritten as \( \sim A \lor B \). Therefore, we can rewrite the given statement:

\[ (~q \implies \sim p) \lor (~q \implies p) = (\sim (~q) \lor \sim p) \lor (\sim (~q) \lor p) \]

### Step 2: Simplifying Negations
The negation of \( ~q \) is \( q \). Thus, we can simplify the expression:

\[ (q \lor \sim p) \lor (q \lor p) \]

### Step 3: Apply Associative and Commutative Laws
Using the associative and commutative properties of logical disjunction, we can rearrange the terms:

\[ q \lor \sim p \lor q \lor p \]

### Step 4: Combine Like Terms
Since \( q \lor q \) is simply \( q \), we can simplify further:

\[ q \lor \sim p \lor p \]

### Step 5: Apply the Law of Excluded Middle
According to the law of excluded middle, \( p \lor \sim p \) is always true (T):

\[ q \lor T \]

### Step 6: Final Simplification
Since \( q \lor T \) is always true, we conclude that the original statement is a tautology:

\[ \text{Tautology} \]

### Conclusion
Thus, the logical statement \( (~q \implies \sim p) \lor (~q \implies p) \) is equivalent to a tautology.