The dual of the converse of the inverse of th | vedclass

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(A) Step $1$: Given statement is $S = p 	o (q 	o r)$.Step $2$: The inverse of $S$ is $∼ p 	o ∼ (q 	o r)$.Step $3$: The converse of the inverse is

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$∼ [p \lor (r 	o q)]$

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(A) Step $1$: Given statement is $S = p 	o (q 	o r)$.Step $2$: The inverse of $S$ is $∼ p 	o ∼ (q 	o r)$.Step $3$: The converse of the inverse is $∼ (q 	o r) 	o ∼ p$.Step $4$: Using $q 	o r \equiv ∼ q \lor r$, the expression becomes $∼ (∼ q \lor r) 	o ∼ p$, which is $(q \land ∼ r) 	o ∼ p$.Step $5$: The dual of a statement is obtained by replacing $\land$ with $\lor$, $\lor$ with $\land$, $T$ with $F$, and $F$ with $T$. The implication $A 	o B$ is $∼ A \lor B$. Thus, $(q \land ∼ r) 	o ∼ p$ is $∼ (q \land ∼ r) \lor ∼ p \equiv (∼ q \lor r) \lor ∼ p$.Step $6$: The dual of $(∼ q \lor r) \lor ∼ p$ is $(∼ q \land r) \land ∼ p$.Step $7$: Alternatively, applying the dual operation directly to the logical form: The dual of $A 	o B$ (i.e., $∼ A \lor B$) is $∼ A \land B$. Applying this to $(q \land ∼ r) 	o ∼ p$ gives $∼ (q \land ∼ r) \land ∼ p \equiv (∼ q \lor r) \land ∼ p$. Checking the options, the expression $∼ [p \lor (r 	o q)]$ is $∼ [p \lor (∼ r \lor q)] \equiv ∼ p \land (r \land ∼ q)$, which matches the dual form.

The dual of the converse of the inverse of the logical statement $p \to (q \to r)$ is equivalent to...

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