The proposition ` ~p vv( p ^^ ~ q)` is equivalent to

Options

A. ` p vv ~ q`
B. ` p to ~ q`
C. ` q to p`
D. ` p ^^ ~ q`

Explanation

To determine the equivalence of the proposition \( \sim p \lor (p \land \sim q) \), we will simplify it step by step.

### Step 1: Rewrite the proposition
The given proposition is:
\[ \sim p \lor (p \land \sim q) \]

### Step 2: Apply the Distributive Law
Using the distributive law of logic, we can rewrite the expression:
\[ \sim p \lor (p \land \sim q) = (\sim p \lor p) \land (\sim p \lor \sim q) \]

### Step 3: Simplify using the Law of Excluded Middle
The expression \( \sim p \lor p \) is always true (T) because either \( p \) is true or \( \sim p \) is true. Thus, we can simplify:
\[ (\sim p \lor p) \land (\sim p \lor \sim q) = T \land (\sim p \lor \sim q) \]

### Step 4: Simplify further
Since \( T \land A = A \) for any proposition \( A \), we have:
\[ T \land (\sim p \lor \sim q) = \sim p \lor \sim q \]

### Step 5: Recognize the equivalence
Now we can express \( \sim p \lor \sim q \) in terms of implications:
\[ \sim p \lor \sim q \equiv p \implies \sim q \]

### Conclusion
Thus, the original proposition \( \sim p \lor (p \land \sim q) \) is equivalent to \( p \implies \sim q \).

### Final Answer
The proposition \( \sim p \lor (p \land \sim q) \) is equivalent to \( p \implies \sim q \).

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