$$\text { Let } p \text { represent } 7

Question

$$\text { Let } p \text { represent } 7 < 8 \text {, } q \text { represent } 2 < 5 < 6 \text {, and } \mathbf { r } \text { represent } 3 < 2 \text {. Decide whether the statement is true or false. }$$
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$\sim ( \sim p  \wedge \sim q ) \wedge  (\sim r \wedge \sim q )$

Quizplus · Accepted Answer

The statement $p$ is true because $7 < 8$ is true, $q$ is true because $2 < 5 < 6$ is true, and $r$ is false because $3 < 2$ is false. Thus, $\sim p$ and $\sim q$ are false, making $\sim p \wedge \sim q$ false. Since the negation of false is true, $\sim (\sim p \wedge \sim q)$ is true. However, $\sim r$ is true (since $r$ is false), but since $q$ is true, $\sim q$ is false, making $(\sim r \wedge \sim q)$ false. Therefore, the entire statement, being a conjunction where one part is true and the other is false, is false.

- ∼(∼p∧∼q)∧(∼r∧∼q)\sim ( \sim p \wedge \sim q ) \wedge (\sim r \wedge \sim q )∼(∼p∧∼q)∧(∼r∧∼q)

- $\sim ( \sim p \wedge \sim q ) \wedge (\sim r \wedge \sim q )$