Question

Which of the following is a tautology?

• A. (P v Q)↔(P v (P↔(Q↔R)))
• B. (P v Q)↔(Q v (P↔(Q↔R)))
• C. (P v Q)↔(P v (Q↔(R↔Q)))
• D. (P v Q)↔(Q v (Q↔(R↔Q)))

Answer 1

The Correct Option is C, D

A tautology is a statement that is always true regardless of the truth values of its individual propositions. Let's evaluate the given options:

1. \(( (P \vee Q) \leftrightarrow (P \vee (P \leftrightarrow (Q \leftrightarrow R))) )\)- When P is true, regardless of the values of Q and R, the left side (P v Q) is true. For the right side, if P is true, then \(( P \vee (P \leftrightarrow (Q \leftrightarrow R)) )\) is always true regardless of the biconditional. - When P is false and Q is true, the left side is still true. On the right side, the biconditional's truth value will determine the overall truth value, making it not always true. This is not a tautology. 
2. \(( (P \vee Q) \leftrightarrow (Q \vee (P \leftrightarrow (Q \leftrightarrow R))) )\) - When Q is true, the whole statement is true regardless of P and R. - When Q is false, and P is true, the biconditional's value will determine the truth of the overall statement. This is not a tautology. 
3. \(( (P \vee Q) \leftrightarrow (P \vee (Q \leftrightarrow (R \leftrightarrow Q))) )\)- When P is true, the whole statement is true regardless of Q and R. - When P is false and Q is true, the biconditional\(( (Q \leftrightarrow (R \leftrightarrow Q)) )\) is always true because Q biconditional with anything involving Q will always be true. This is a tautology.
4. \(( (P \vee Q) \leftrightarrow (Q \vee (Q \leftrightarrow (R \leftrightarrow Q))) )\)- Similar to the 3rd option, if Q is true, then the statement is always true regardless of P and R. This is a tautology.

Therefore, the tautologies are: 3. \(( (P v Q) leftrightarrow (P v (Q \leftrightarrow (R \leftrightarrow Q))) )\) 4. \(( (P v Q) \leftrightarrow (Q v (Q \leftrightarrow (R \leftrightarrow Q)))\)