The compound statement `(phArr q)vv(p hArr ~q)` is logically equivalent to

Options

A. `phArrq`
B. `pvvq`
C. tautology
D. contradiction

Answer

To determine the logical equivalence of the compound statement \( (p \Rightarrow q) \lor (p \Rightarrow \neg q) \), we will follow these steps:

### Step 1: Understand the Implications
The statement \( p \Rightarrow q \) can be rewritten using logical equivalences:
\[ p \Rightarrow q \equiv \neg p \lor q \]
Similarly, for \( p \Rightarrow \neg q \):
\[ p \Rightarrow \neg q \equiv \neg p \lor \neg q \]

### Step 2: Rewrite the Compound Statement
Now, we can rewrite the original compound statement:
\[ (p \Rightarrow q) \lor (p \Rightarrow \neg q) \equiv (\neg p \lor q) \lor (\neg p \lor \neg q) \]

### Step 3: Apply Distributive Law
Using the distributive law of logic, we can combine the two parts:
\[ (\neg p \lor q) \lor (\neg p \lor \neg q) \equiv \neg p \lor (q \lor \neg q) \]

### Step 4: Simplify Using Tautology
The expression \( q \lor \neg q \) is a tautology (always true):
\[ q \lor \neg q \equiv \text{True} \]
Thus, we can simplify further:
\[ \neg p \lor \text{True} \equiv \text{True} \]

### Step 5: Conclusion
Since the entire expression simplifies to True, we conclude that:
\[ (p \Rightarrow q) \lor (p \Rightarrow \neg q) \text{ is a tautology.} \]

### Final Answer
The compound statement \( (p \Rightarrow q) \lor (p \Rightarrow \neg q) \) is logically equivalent to a tautology.