Which of the following statements is NOT logi | vedclass

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Question

(A) First,analyze the given statement: $(p 	o \sim p) 	o (p 	o q)$.Using the equivalence $A 	o B \equiv \sim A \lor B$,we get $\sim (p 	o \sim p)

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$(p 	o p) 	o (p 	o \sim p)$

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(A) First,analyze the given statement: $(p 	o \sim p) 	o (p 	o q)$.Using the equivalence $A 	o B \equiv \sim A \lor B$,we get $\sim (p 	o \sim p) \lor (\sim p \lor q)$.Since $\sim (p 	o \sim p) \equiv \sim (\sim p \lor \sim p) \equiv \sim (\sim p) \equiv p$,the expression simplifies to $p \lor \sim p \lor q$,which is a tautology $(T)$.Now,check the options:Option $A$: $(p 	o p) 	o (p 	o \sim p) \equiv T 	o (\sim p \lor \sim p) \equiv \sim p \lor \sim p \equiv \sim p$. This is not a tautology.Option $B$: $q 	o (p 	o q) \equiv \sim q \lor (\sim p \lor q) \equiv (\sim q \lor q) \lor \sim p \equiv T \lor \sim p \equiv T$.Option $C$: $(q 	o \sim p) 	o (q 	o p) \equiv \sim (\sim q \lor \sim p) \lor (\sim q \lor p) \equiv (q \land p) \lor \sim q \lor p \equiv (q \lor \sim q) \land (p \lor \sim q) \lor p \equiv T \land (p \lor \sim q \lor p) \equiv p \lor \sim q$. This is not a tautology.Since the original statement is a tautology and options $A$ and $C$ are not,the question implies finding which is not equivalent. Given the structure,$A$ is clearly not equivalent.

Which of the following statements is $NOT$ logically equivalent to $(p \to \sim p) \to (p \to q)$?

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