The compound statement `(prarrq)vv (p^^~q)` is logically equivalent to

Options

A. `pharr q`
B. `p ^^ q`
C. tautology
D. contradiction

Answer

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

### Step 1: Understand the components of the statement
The statement consists of two parts:
1. \( p \rightarrow q \) (p implies q)
2. \( p \land \neg q \) (p and not q)

### Step 2: Create a truth table
We will create a truth table to evaluate the truth values of the compound statement based on all possible truth values of \( p \) and \( q \).

| \( p \) | \( q \) | \( \neg q \) | \( p \rightarrow q \) | \( p \land \neg q \) | \( (p \rightarrow q) \lor (p \land \neg q) \) |
|---------|---------|---------------|------------------------|-----------------------|-----------------------------------------------|
| T | T | F | T | F | T |
| T | F | T | F | T | T |
| F | T | F | T | F | T |
| F | F | T | T | F | T |

### Step 3: Evaluate each column
- **Column for \( \neg q \)**: The negation of \( q \) flips the truth value of \( q \).
- **Column for \( p \rightarrow q \)**: This is true unless \( p \) is true and \( q \) is false.
- **Column for \( p \land \neg q \)**: This is true only when both \( p \) is true and \( \neg q \) is true (which means \( q \) is false).
- **Final column**: This column combines the results of the previous two columns using logical OR (\( \lor \)). It is true if at least one of the statements is true.

### Step 4: Analyze the final column
From the truth table, we can see that the final column is true for all combinations of \( p \) and \( q \). Therefore, the expression \( (p \rightarrow q) \lor (p \land \neg q) \) is a tautology.

### Conclusion
The compound statement \( (p \rightarrow q) \lor (p \land \neg q) \) is logically equivalent to a tautology.