If p,q,r have truth values T,F,T respectively, which of the following is true ?

Options

A. ` ( p to q) ^^ r`
B. ` ( p to q) ^^ ~ r`
C. ` ( p^^ q) ^^ ( p vv r)`
D. ` q to ( p ^^ r)`

Answer

To determine which of the given statements is true based on the truth values of \( p \), \( q \), and \( r \) (where \( p = T \), \( q = F \), and \( r = T \)), we will evaluate each option step by step.

### Step 1: Evaluate \( P \implies Q \) and \( R \)

1. **Identify the values**:
- \( p = T \)
- \( q = F \)
- \( r = T \)

2. **Evaluate \( P \implies Q \)**:
- The implication \( P \implies Q \) is only false when \( p \) is true and \( q \) is false.
- Here, \( T \implies F \) is false.

3. **Combine with \( R \)**:
- Now we have \( (P \implies Q) \land R \).
- This becomes \( F \land T \).
- Since false and true is false, the result is **false**.

### Step 2: Evaluate \( P \implies Q \) and \( \neg R \)

1. **Evaluate \( \neg R \)**:
- Since \( r = T \), \( \neg R = F \).

2. **Combine**:
- We have \( (P \implies Q) \land \neg R \).
- This becomes \( F \land F \).
- The result is **false**.

### Step 3: Evaluate \( Q \implies P \) and \( R \)

1. **Evaluate \( Q \implies P \)**:
- Here, \( q = F \) and \( p = T \).
- The implication \( Q \implies P \) is true because false implies true is true.

2. **Combine with \( R \)**:
- We have \( (Q \implies P) \land R \).
- This becomes \( T \land T \).
- The result is **true**.

### Step 4: Evaluate \( P \land Q \) and \( P \lor R \)

1. **Evaluate \( P \land Q \)**:
- Here, \( p = T \) and \( q = F \).
- The conjunction \( P \land Q \) is \( T \land F \), which is **false**.

2. **Evaluate \( P \lor R \)**:
- Here, \( p = T \) and \( r = T \).
- The disjunction \( P \lor R \) is \( T \lor T \), which is **true**.

3. **Combine**:
- We have \( (P \land Q) \lor (P \lor R) \).
- This becomes \( F \lor T \), which is **true**.

### Conclusion

After evaluating all options, we find that:

- Option 1: False
- Option 2: False
- Option 3: False
- Option 4: True

Thus, **the correct answer is Option 4**.