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

Options

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

Answer

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

### Step 1: Understand the components of the statement
The statement consists of two main parts:
1. \( p \land q \) (p AND q)
2. \( \neg q \rightarrow r \) (NOT q implies r)

### Step 2: Analyze the implication
The implication \( \neg q \rightarrow r \) can be rewritten using the equivalence:
\[ A \rightarrow B \equiv \neg A \lor B \]
Thus, we can rewrite \( \neg q \rightarrow r \) as:
\[ \neg (\neg q) \lor r \equiv q \lor r \]

### Step 3: Rewrite the original statement
Now, substituting back into the original statement, we have:
\[ (p \land q) \lor (q \lor r) \]

### Step 4: Simplify the expression
This can be further simplified using the associative and commutative properties of logical operations:
\[ (p \land q) \lor (q \lor r) = (p \land q) \lor q \lor r \]
Since \( q \lor q \) is just \( q \), we can simplify it to:
\[ (p \land q) \lor r \]

### Step 5: Evaluate with \( p \) being true
Given that \( p \) is true, we now consider the truth values of \( q \) and \( r \):
1. If \( q \) is true, then \( p \land q \) is true, making the whole expression true regardless of \( r \).
2. If \( q \) is false, then \( p \land q \) is false. The truth value of the expression now depends on \( r \):
- If \( r \) is true, the expression is true.
- If \( r \) is false, the expression is false.

### Step 6: Conclusion
Thus, the truth value of the statement \( (p \land q) \lor (\neg q \rightarrow r) \) depends on the values of \( q \) and \( r \):
- The expression is true if \( q \) is true or if \( r \) is true when \( q \) is false.
- The expression is false only when \( q \) is false and \( r \) is also false.

### Final Truth Value
Therefore, the truth value of the statement can be either true or false depending on the values of \( q \) and \( r \).

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