If `(p ^^ ~r) implies (q vv r)` is false and q and r are both false, then p is

Options

A. true
B. false
C. may be true or false
D. data insufficient

Answer

To solve the problem step by step, we need to analyze the logical expression given and the truth values of the variables involved.

### Step 1: Understand the given expression
The expression we need to analyze is:
\[ (p \land \neg r) \implies (q \lor r) \]
We know that this expression is false.

### Step 2: Identify the truth values of q and r
We are given that both \( q \) and \( r \) are false:
\[ q = \text{False}, \quad r = \text{False} \]

### Step 3: Evaluate the right-hand side of the implication
Now, let's evaluate \( q \lor r \):
\[ q \lor r = \text{False} \lor \text{False} = \text{False} \]

### Step 4: Analyze the implication
The implication \( (p \land \neg r) \implies (q \lor r) \) is false only when the left-hand side is true and the right-hand side is false. Since we have already determined that the right-hand side \( (q \lor r) \) is false, we need the left-hand side \( (p \land \neg r) \) to be true.

### Step 5: Evaluate the left-hand side
Now let's evaluate \( \neg r \):
\[ \neg r = \neg \text{False} = \text{True} \]
Thus, the left-hand side becomes:
\[ p \land \neg r = p \land \text{True} = p \]
For the left-hand side \( (p \land \neg r) \) to be true, \( p \) must be true.

### Conclusion
Therefore, we conclude that:
\[ p = \text{True} \]

### Final Answer
Thus, the value of \( p \) is **True**.

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