In propositional logic \(P \leftrightarrow Q\) is equivalent to (Where \(\sim\) denotes NOT) :
\((\sim P \vee Q) \wedge (\sim Q \vee P)\)
\(P \leftrightarrow Q = (P \rightarrow Q) \wedge (Q \rightarrow P)\)
Using \(X \rightarrow Y = \sim X \vee Y\): \(P \rightarrow Q = \sim P \vee Q\) and \(Q \rightarrow P = \sim Q \vee P\)
\(P \leftrightarrow Q = (\sim P \vee Q) \wedge (\sim Q \vee P)\)
Check: with P true and Q false the biconditional is false, and \((\sim P \vee Q) = \) false, as required. The first form is true only when both are false, the third reduces to \(P \vee Q\), and the fourth is a tautology.
Hence, \((\sim P \vee Q) \wedge (\sim Q \vee P)\).
Similar Questions
Consider the compound propositions given below as :
(a) \(p \vee \sim (p \wedge q)\) (b) \((p \wedge \sim q) \vee \sim (p \wedge q)\)
(c) \(p \wedge (q \vee r)\)
Which of the above propositions are tautologies ?
Match the following :
List - I List - II (a) \((p \to q) \Leftrightarrow (\neg q \to \neg p)\) (i) Contrapositive (b) \([(p \wedge q) \to r] \Leftrightarrow [p \to (q \to r)]\) (ii) Exportation law (c) \((p \to q) \Leftrightarrow [(p \wedge \neg q) \to o]\) (iii) Reductio ad absurdum (d) \((p \leftrightarrow q) \Leftrightarrow [(p \to q) \wedge (q \to p)]\) (iv) Equivalence Codes :
Which of the following is NOT logically equivalent to P→Q?
Consider the following if p and q are two statements.
(A) ~(p∧q) ≡ ~p∨~q
(B) ~(p∨q) ≡ ~p∧~q
(C) p∧~p ≡ T
(D) ~(p⇒q) ≡ p∧~q
(E) p∨q ≡ ~p∨~q
Choose the correct answer from the options given below :
Match the LIST-I with LIST-II: Match the logical equivalence propositions
LIST-I LIST-II A. p → q I. (p ∧ q) ∨ (¬p ∧ ¬q) B. ¬(p ∨ (¬p ∧ q)) II. ¬p ∨ q C. p ↔ q III. ¬(p ∨ q) D. ¬(p ↔ q) IV. ¬p ↔ q Choose the correct answer from the options given below:
Important Questions from Propositional Equivalence
Consider the compound propositions given below as :
(a) \(p \vee \sim (p \wedge q)\) (b) \((p \wedge \sim q) \vee \sim (p \wedge q)\)
(c) \(p \wedge (q \vee r)\)
Which of the above propositions are tautologies ?
Match the following :
List - I List - II (a) \((p \to q) \Leftrightarrow (\neg q \to \neg p)\) (i) Contrapositive (b) \([(p \wedge q) \to r] \Leftrightarrow [p \to (q \to r)]\) (ii) Exportation law (c) \((p \to q) \Leftrightarrow [(p \wedge \neg q) \to o]\) (iii) Reductio ad absurdum (d) \((p \leftrightarrow q) \Leftrightarrow [(p \to q) \wedge (q \to p)]\) (iv) Equivalence Codes :
Which of the following is NOT logically equivalent to P→Q?
Consider the following if p and q are two statements.
(A) ~(p∧q) ≡ ~p∨~q
(B) ~(p∨q) ≡ ~p∧~q
(C) p∧~p ≡ T
(D) ~(p⇒q) ≡ p∧~q
(E) p∨q ≡ ~p∨~q
Choose the correct answer from the options given below :
Match the LIST-I with LIST-II: Match the logical equivalence propositions
LIST-I LIST-II A. p → q I. (p ∧ q) ∨ (¬p ∧ ¬q) B. ¬(p ∨ (¬p ∧ q)) II. ¬p ∨ q C. p ↔ q III. ¬(p ∨ q) D. ¬(p ↔ q) IV. ¬p ↔ q Choose the correct answer from the options given below:
