Q.Using the rules of negation, write the negation of the following, with justification: $(\sim p \vee \sim q) \vee (p \wedge \sim q)$

Negate using De Morgan's law and $\sim(q\to r)\equiv q\wedge\sim r$.

$\sim[p\wedge(q\to r)] \equiv \sim p \vee \sim(q\to r) \equiv \sim p\vee(q\wedge\sim r)$

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Construct the truth table for both sides and compare.

The columns for $\sim(p\vee q)$ and $\sim p \wedge \sim q$ are identical (F, F, F, T) in every row.

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Apply De Morgan's law to $\sim[p\land(q\to r)]$.

We need $\sim[p\land(q\to r)]$.

By De Morgan's law, $\sim(A\land B)\equiv \sim A\lor\sim B$. Here $A=p$, $B=(q\to r)$, so

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

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