Which of the following statements is a tautol | vedclass

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(D) statement is a tautology if its truth value is $T$ for all possible truth values of its components.Let us evaluate option $B$: $p \Rightarrow ((\s

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$q \Rightarrow ((\sim p) \vee q)$

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(D) statement is a tautology if its truth value is $T$ for all possible truth values of its components.Let us evaluate option $B$: $p \Rightarrow ((\sim p) \vee q)$.This is equivalent to $(\sim p) \vee ((\sim p) \vee q)$ by the implication law $A \Rightarrow B \equiv (\sim A) \vee B$.Using the associative law,this becomes $(\sim p \vee \sim p) \vee q$,which simplifies to $(\sim p) \vee q$.Since this is not always $T$ (e.g.,if $p=T, q=F$,then $\sim p \vee q = F$),let us re-examine the options.Actually,option $D$ is $q \Rightarrow ((\sim p) \vee q)$.This is equivalent to $(\sim q) \vee ((\sim p) \vee q) = (\sim q \vee q) \vee (\sim p) = T \vee (\sim p) = T$.Since the result is always $T$,option $D$ is a tautology.

Which of the following statements is a tautology?

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