The statement `(p^^(~q))vv((~p)^^q)vv((~p)^^(~q))` is equalent to ____

Options

A. `(~p)vv(~q)`
B. `pvv(~q)`
C. `pvvq`
D. `(~p)vvq`

Answer

To solve the logical expression \( (p \land \neg q) \lor (\neg p \land q) \lor (\neg p \land \neg q) \), we will simplify it step by step.

### Step 1: Identify the components of the expression
The expression consists of three parts:
1. \( p \land \neg q \)
2. \( \neg p \land q \)
3. \( \neg p \land \neg q \)

### Step 2: Rewrite the expression
We can rewrite the expression for clarity:
\[ (p \land \neg q) \lor (\neg p \land q) \lor (\neg p \land \neg q) \]

### Step 3: Combine terms
Notice that \( \neg p \land q \) and \( \neg p \land \neg q \) both contain \( \neg p \). We can factor out \( \neg p \):
\[ (p \land \neg q) \lor \neg p \land (q \lor \neg q) \]

### Step 4: Simplify using the law of excluded middle
The expression \( q \lor \neg q \) is always true (T), so we can simplify further:
\[ (p \land \neg q) \lor \neg p \]

### Step 5: Analyze the expression
Now we have \( (p \land \neg q) \lor \neg p \). This means:
- If \( p \) is false, the expression is true regardless of \( q \).
- If \( p \) is true, the expression is true if \( q \) is false.

### Step 6: Write the equivalent expression
The simplified expression can be interpreted as \( \neg p \lor (p \land \neg q) \). This is equivalent to:
\[ \neg p \lor \neg q \]

### Final Result
Thus, the original statement \( (p \land \neg q) \lor (\neg p \land q) \lor (\neg p \land \neg q) \) is equivalent to:
\[ \neg p \lor \neg q \]

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