The statement `~prarr(qrarrp)` is equivalent to

Options

A. `prarr(prarrq)`
B. `prarr(p vv q)`
C. `q rarrp`
D. `q rarr (prarrq)`

Answer

To solve the problem of determining the equivalence of the statement `~p → (q → p)`, we can follow these steps:

### Step 1: Understand the Statement
The statement `~p → (q → p)` can be interpreted as "if not p, then (if q then p)". We need to analyze this statement logically.

### Step 2: Rewrite the Implications
Recall that the implication \( A \to B \) can be rewritten as \( \neg A \lor B \). Thus, we can rewrite the statement:
- \( q \to p \) can be rewritten as \( \neg q \lor p \).
- Therefore, \( ~p → (q → p) \) becomes \( \neg p \to (\neg q \lor p) \).

### Step 3: Rewrite the Negation
Now, applying the implication transformation to \( \neg p \to (\neg q \lor p) \):
- This becomes \( \neg(\neg p) \lor (\neg q \lor p) \).
- Simplifying \( \neg(\neg p) \) gives us \( p \), so we have \( p \lor (\neg q \lor p) \).

### Step 4: Simplify the Expression
Using the associative and commutative properties of logical disjunction (OR):
- We can simplify \( p \lor (\neg q \lor p) \) to \( p \lor \neg q \).

### Step 5: Final Result
Thus, the original statement `~p → (q → p)` simplifies to \( p \lor \neg q \).

### Conclusion
The statement `~p → (q → p)` is equivalent to \( p \lor \neg q \).