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Mathematics Question on Differential equations

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

(P v Q)↔(P v (Q↔(R↔Q)))

Explanation

Solution

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∨Q)↔(P∨(P↔(Q↔R))))( (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∨(P↔(Q↔R)))( 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∨Q)↔(Q∨(P↔(Q↔R))))( (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∨Q)↔(P∨(Q↔(R↔Q))))( (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↔(R↔Q)))( (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∨Q)↔(Q∨(Q↔(R↔Q))))( (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. ((PvQ)leftrightarrow(Pv(Q↔(R↔Q))))( (P v Q) leftrightarrow (P v (Q \leftrightarrow (R \leftrightarrow Q))) ) 4. ((PvQ)↔(Qv(Q↔(R↔Q)))( (P v Q) \leftrightarrow (Q v (Q \leftrightarrow (R \leftrightarrow Q)))