If p, q, r are substatements with truth values T, T, F then the Statemetn `r to (p^^~q) vv (~q^^~r)` will be

Options

A. True
B. Flase
C. may be true or false
D. None of these

Answer

To solve the statement \( r \to (p \land \neg q) \lor (\neg q \land \neg r) \) given the truth values \( p = T \), \( q = T \), and \( r = F \), we will follow these steps:

### Step 1: Identify the negations of the statements
- Since \( p \) is true (T), \( \neg p \) will be false (F).
- Since \( q \) is true (T), \( \neg q \) will be false (F).
- Since \( r \) is false (F), \( \neg r \) will be true (T).

### Step 2: Substitute the truth values into the statement
The statement can be rewritten as:
\[ r \to (p \land \neg q) \lor (\neg q \land \neg r) \]
Substituting the truth values:
\[ F \to (T \land F) \lor (F \land T) \]

### Step 3: Evaluate the components of the statement
1. Evaluate \( p \land \neg q \):
- \( T \land F = F \)

2. Evaluate \( \neg q \land \neg r \):
- \( F \land T = F \)

Now the statement simplifies to:
\[ F \to F \lor F \]

### Step 4: Evaluate the disjunction
Now we evaluate the disjunction:
\[ F \lor F = F \]

### Step 5: Evaluate the implication
Now we have:
\[ F \to F \]
In logic, an implication \( A \to B \) is true unless \( A \) is true and \( B \) is false. Since both \( A \) and \( B \) are false, the implication is true:
\[ F \to F = T \]

### Final Answer
Thus, the statement \( r \to (p \land \neg q) \lor (\neg q \land \neg r) \) evaluates to **True**.

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