## Thinking Mathematically (6th Edition)

The negation of the statement $p\wedge \left( r\to \sim s \right)$ using De Morgan’s law is given as $\sim p\vee \sim \left( r\to \sim s \right)$. To form the negation of a conditional statement, leave the antecedent (the first part unchanged, change the $\text{if-then}$ connective to and) and negate the consequent (the second part). Therefore, $\sim \left( r\to \sim s \right)\equiv r\wedge \sim \left( \sim s \right)$. As$\sim \left( \sim s \right)=s$, therefore, $r\wedge \sim \left( \sim s \right)\equiv r\wedge s$. Thus, the negation of $p\wedge \left( r\to \sim s \right)$ is $\sim p\vee \left( r\wedge s \right)$.