Statement `(P implies Q) vv(R implies Q)` Is logically equivalent to

Options

A. `(P implies R) vv (Q implies R)`
B. `(PvvR)impliesQ`
C. `(P implies R)^^(Q implies R)`
D. `(P^^R)impliesQ`

Answer

To determine the logical equivalence of the statement \( (P \implies Q) \lor (R \implies Q) \), we can follow these steps:

### Step 1: Rewrite the implications
The implication \( P \implies Q \) can be rewritten using logical equivalences. Recall that \( P \implies Q \) is equivalent to \( \neg P \lor Q \). Similarly, \( R \implies Q \) can be rewritten as \( \neg R \lor Q \).

Thus, we can rewrite the original statement:
\[ (P \implies Q) \lor (R \implies Q) = (\neg P \lor Q) \lor (\neg R \lor Q) \]

### Step 2: Apply the Associative Law
Using the associative law of disjunction, we can rearrange the terms:
\[ (\neg P \lor Q) \lor (\neg R \lor Q) = \neg P \lor \neg R \lor Q \]

### Step 3: Factor out common terms
Notice that \( Q \) appears in both parts of the disjunction. We can factor it out:
\[ \neg P \lor \neg R \lor Q = Q \lor (\neg P \lor \neg R) \]

### Step 4: Rewrite using implications
The expression \( \neg P \lor \neg R \) can be rewritten as \( \neg (P \land R) \) using De Morgan's laws. Therefore, we have:
\[ Q \lor \neg (P \land R) \]

### Step 5: Final implication form
This can be interpreted as:
\[ \neg (P \land R) \lor Q \]
This is equivalent to:
\[ (P \land R) \implies Q \]

### Conclusion
Thus, the original statement \( (P \implies Q) \lor (R \implies Q) \) is logically equivalent to:
\[ (P \land R) \implies Q \]