Let p and q be propositions- Using only the truth table decide whether p- q dose not imply p- q is true or false-

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Question

Let p and q be propositions. Using only the truth table decide whether $p \Leftrightarrow q$ dose not imply $p \to⇁ q$ is true or false.

False

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True

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Solution

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(p \land \sim q) \leftrightarrow (p \to q)

p \Rightarrow\sim (p \land \sim q)

I f t h e p r o p o s i t i o n \neg p \Rightarrow q i s t r u e,

\neg P \lor (p \Rightarrow q) w h e r e, \neg i s n e g a t i o n,^{'} \lor^{'}

\Rightarrow

\sim [(p \lor \sim q) \to (p \land \sim q)] \equiv (p \lor \sim q) \land (\sim \lor q)

p \leftrightarrow q \equiv (p \land q) \lor (\sim p \land \sim q)

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