Given three statements P: 5 is a prime number, Q:7 is a factor of 192, R:The LCM of 5 & 7 is 35 Then which of the following statements are true (a) `Pv(~Q^^R)` (b) `~P^^(~Q^^R)` (c) `(PvQ)^^~R` (d) `~P^^(~Q^^R)`

Options

A. `(p wedgeq)vee (~r)`
B. `(~p)wedge(~q wedge r)`
C. `(~p) vee (q wedge r)`
D. `p vee(~q wedge r)`

Answer

To solve the problem, we need to evaluate the truth values of the statements P, Q, and R, and then analyze the given options based on these truth values.

### Step 1: Determine the truth values of P, Q, and R
1. **Statement P**: "5 is a prime number."
- This statement is **True** (T).

2. **Statement Q**: "7 is a factor of 192."
- To check this, we divide 192 by 7.
- \(192 \div 7 = 27.42857\) (not an integer)
- Therefore, this statement is **False** (F).

3. **Statement R**: "The LCM of 5 and 7 is 35."
- The LCM of two prime numbers is their product.
- \(LCM(5, 7) = 5 \times 7 = 35\)
- Hence, this statement is **True** (T).

### Step 2: Summary of truth values
- P: True (T)
- Q: False (F)
- R: True (T)

### Step 3: Evaluate the options
Now we will evaluate each option using the truth values we have determined.

**Option (a)**: \(P \lor (\neg Q \land R)\)
- \(\neg Q\) (not Q) is True (since Q is False).
- Therefore, \(\neg Q \land R\) is \(T \land T = T\).
- Now, \(P \lor T = T \lor T = T\).
- **Result**: True (T)

**Option (b)**: \(\neg P \land (\neg Q \land R)\)
- \(\neg P\) is False (since P is True).
- Therefore, \(\neg P \land (\neg Q \land R) = F \land T = F\).
- **Result**: False (F)

**Option (c)**: \((P \lor Q) \land \neg R\)
- \(P \lor Q = T \lor F = T\).
- \(\neg R\) is False (since R is True).
- Therefore, \(T \land F = F\).
- **Result**: False (F)

**Option (d)**: \(\neg P \land (\neg Q \land R)\)
- This is the same as option (b), which we already evaluated.
- **Result**: False (F)

### Conclusion
The only true statement among the options is:
- **Option (a)**: \(P \lor (\neg Q \land R)\) is **True**.

### Final Answer
The true statement is (a).