If `(p ^^ ~q) vv (q ^^r)` is true and q and r both true then p is-

Options

A. True
B. False
C. may be true or false
D. none

Answer

To solve the problem, we need to analyze the logical expression given and the conditions stated. Let's break it down step by step.

### Step 1: Understand the given expression
The expression we have is:
\[ (p \land \neg q) \lor (q \land r) \]
We know that this entire expression is true.

### Step 2: Analyze the conditions
We are given that both \( q \) and \( r \) are true:
\[ q = \text{True}, \quad r = \text{True} \]

### Step 3: Evaluate the second part of the expression
Now, let's evaluate the second part of the expression \( (q \land r) \):
\[ (q \land r) = (\text{True} \land \text{True}) = \text{True} \]

### Step 4: Determine the overall truth of the expression
Since the expression is a disjunction (OR), it will be true if at least one of the components is true. We already know that:
\[ (q \land r) = \text{True} \]
Thus, the entire expression:
\[ (p \land \neg q) \lor (q \land r) = \text{True} \]
is satisfied regardless of the truth value of \( (p \land \neg q) \).

### Step 5: Analyze the first part of the expression
Next, we need to consider \( (p \land \neg q) \):
Since \( q \) is true, \( \neg q \) (not q) will be false:
\[ \neg q = \text{False} \]
Thus, \( (p \land \neg q) \) becomes:
\[ (p \land \text{False}) = \text{False} \]
This means that the first part of the expression does not contribute to the truth of the overall expression.

### Step 6: Conclusion about \( p \)
Since the overall expression is true due to the second part \( (q \land r) \), we cannot determine the truth value of \( p \) from this expression alone. Therefore, \( p \) can be either true or false.

### Final Answer
Thus, \( p \) can be either true or false.

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