Which of the following is logically equivalent to `~(~prArr q)`?

Options

A. Tautology and contradiction
B. Neither tautology nor contradiction
C. Contradiction
D. Tautology

Answer

To solve the problem of determining which of the following is logically equivalent to `~(~p → q)`, we can follow these steps:

### Step 1: Understand the expression
The expression `~(~p → q)` consists of a negation of an implication. We need to simplify this expression to find its logical equivalence.

### Step 2: Rewrite the implication
Recall that the implication `A → B` can be rewritten using logical operators:
\[ A → B \equiv ~A ∨ B \]
Thus, we can rewrite `~p → q` as:
\[ ~p → q \equiv ~~p ∨ q \]
So, we have:
\[ ~p → q \equiv p ∨ q \]

### Step 3: Apply negation
Now we apply the negation to the rewritten expression:
\[ ~(~p → q) \equiv ~(p ∨ q) \]

### Step 4: Use De Morgan's Law
According to De Morgan's Law, the negation of a disjunction is the conjunction of the negations:
\[ ~(p ∨ q) \equiv ~p ∧ ~q \]

### Step 5: Conclusion
Thus, the expression `~(~p → q)` simplifies to `~p ∧ ~q`.

### Final Answer
The logically equivalent expression to `~(~p → q)` is `~p ∧ ~q`.