If p, q and r are three logical statements then the truth value of the statement `(p^^~q)vv(qrarr r)`, where p is true, is

Options

A. True if q is true
B. False if q is true
C. True if q is false
D. False is q is false

Answer

To determine the truth value of the logical statement \( (p \land \neg q) \lor (q \rightarrow r) \) given that \( p \) is true, we can follow these steps:

### Step 1: Assign the truth value to \( p \)
Since it is given that \( p \) is true, we can write:
\[ p = \text{True} \]

### Step 2: Analyze \( \neg q \)
The negation of \( q \) will depend on the truth value of \( q \). Therefore, we will consider two cases for \( q \): when \( q \) is true and when \( q \) is false.

### Step 3: Case 1 - Assume \( q \) is true
If \( q \) is true:
\[ q = \text{True} \]
Then:
\[ \neg q = \text{False} \]

Now, substitute these values into the expression:
\[ (p \land \neg q) = (\text{True} \land \text{False}) = \text{False} \]

Next, we need to evaluate \( (q \rightarrow r) \):
The implication \( q \rightarrow r \) is equivalent to \( \neg q \lor r \). Since \( q \) is true, we can write:
\[ q \rightarrow r = \text{True} \rightarrow r \]
This is true if \( r \) is true and false if \( r \) is false.

So, we have:
- If \( r \) is true, \( (q \rightarrow r) = \text{True} \)
- If \( r \) is false, \( (q \rightarrow r) = \text{False} \)

Now, substitute back into the original expression:
\[ (p \land \neg q) \lor (q \rightarrow r) = \text{False} \lor (q \rightarrow r) \]

- If \( r \) is true:
\[ \text{False} \lor \text{True} = \text{True} \]
- If \( r \) is false:
\[ \text{False} \lor \text{False} = \text{False} \]

### Step 4: Case 2 - Assume \( q \) is false
If \( q \) is false:
\[ q = \text{False} \]
Then:
\[ \neg q = \text{True} \]

Now substitute these values into the expression:
\[ (p \land \neg q) = (\text{True} \land \text{True}) = \text{True} \]

Next, evaluate \( (q \rightarrow r) \):
Since \( q \) is false:
\[ q \rightarrow r = \text{False} \rightarrow r \]
This is always true regardless of the truth value of \( r \).

So, we have:
\[ (p \land \neg q) \lor (q \rightarrow r) = \text{True} \lor \text{True} = \text{True} \]

### Conclusion
From the above analysis:
- If \( q \) is true, the truth value depends on \( r \) (either true or false).
- If \( q \) is false, the expression is always true.

Thus, the overall truth value of the statement \( (p \land \neg q) \lor (q \rightarrow r) \) is **True** if \( q \) is false, and it can be either true or false if \( q \) is true, depending on \( r \).

### Final Answer
The truth value of the statement is **True** if \( q \) is false.