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

Options

A. `phArr q`
B. `p^^q`
C. Tautology
D. Contradication

Answer

To determine the logical equivalence of the compound statement \((p \rightarrow \neg q) \lor (p \land q)\), we will construct a truth table and analyze the results step by step.

### Step 1: Set Up the Truth Table
We will create a truth table with the following columns:
- \(P\)
- \(Q\)
- \(\neg Q\) (negation of \(Q\))
- \(P \rightarrow \neg Q\) (implication)
- \(P \land Q\) (conjunction)
- \((P \rightarrow \neg Q) \lor (P \land Q)\) (disjunction)

### Step 2: Fill in the Values for \(P\) and \(Q\)
The possible truth values for \(P\) and \(Q\) are:
- True (T)
- False (F)

Thus, we will have the following combinations:
1. \(P = T, Q = T\)
2. \(P = T, Q = F\)
3. \(P = F, Q = T\)
4. \(P = F, Q = F\)

### Step 3: Calculate \(\neg Q\)
- If \(Q = T\), then \(\neg Q = F\)
- If \(Q = F\), then \(\neg Q = T\)

So, the values for \(\neg Q\) will be:
1. \(F\)
2. \(T\)
3. \(F\)
4. \(T\)

### Step 4: Calculate \(P \rightarrow \neg Q\)
The implication \(P \rightarrow \neg Q\) is defined as:
- True unless \(P\) is true and \(\neg Q\) is false.

Calculating this:
1. \(T \rightarrow F = F\)
2. \(T \rightarrow T = T\)
3. \(F \rightarrow F = T\)
4. \(F \rightarrow T = T\)

So, the values for \(P \rightarrow \neg Q\) will be:
1. \(F\)
2. \(T\)
3. \(T\)
4. \(T\)

### Step 5: Calculate \(P \land Q\)
The conjunction \(P \land Q\) is true only if both \(P\) and \(Q\) are true:
1. \(T \land T = T\)
2. \(T \land F = F\)
3. \(F \land T = F\)
4. \(F \land F = F\)

So, the values for \(P \land Q\) will be:
1. \(T\)
2. \(F\)
3. \(F\)
4. \(F\)

### Step 6: Calculate \((P \rightarrow \neg Q) \lor (P \land Q)\)
Now we take the disjunction of the previous two results:
1. \(F \lor T = T\)
2. \(T \lor F = T\)
3. \(T \lor F = T\)
4. \(T \lor F = T\)

So, the final values for \((P \rightarrow \neg Q) \lor (P \land Q)\) will be:
1. \(T\)
2. \(T\)
3. \(T\)
4. \(T\)

### Conclusion
Since all entries in the final column are true, the compound statement \((p \rightarrow \neg q) \lor (p \land q)\) is a tautology. Therefore, it is logically equivalent to **True**.