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Question 70

The negation of the Boolean expression $$\sim q \wedge p \Rightarrow \sim p \vee q$$ is logically equivalent to

Given statement,

$$\sim q\land p\Rightarrow \sim p\lor q$$

Now,

$$\sim p\lor q\equiv p\Rightarrow q$$

and

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

Therefore, the statement becomes

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

Let

$$A=(p\Rightarrow q)$$

Then the statement is

$$\sim A\Rightarrow A$$

The negation of

$$\sim A\Rightarrow A$$

is

$$\sim A\land \sim A$$

$$=\sim A$$

Substituting back,

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

Hence, the required negation is $$\boxed{\sim(p\Rightarrow q)}$$.

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