determine which, if any, of the three statements are equivalent. i) jan is well or jan is still recovering…

determine which, if any, of the three statements are equivalent. i) jan is well or jan is still recovering. ii) if jan is still recovering, then jan is not well. iii) if jan is well, then jan is not still recovering. a. i and ii are equivalent b. ii and iii are equivalent c. i, ii, and iii are equivalent d. none are equivalent

determine which, if any, of the three statements are equivalent. i) jan is well or jan is still recovering. ii) if jan is still recovering, then jan is not well. iii) if jan is well, then jan is not still recovering. a. i and ii are equivalent b. ii and iii are equivalent c. i, ii, and iii are equivalent d. none are equivalent

Answer

Explanation:

Step1: Define variables

Let $p$: Jan is well, $q$: Jan is still recovering. Statement I: $p\vee q$. Statement II: $q\rightarrow\neg p$. Statement III: $p\rightarrow\neg q$.

Step2: Use logical - equivalence rules

Recall that $q\rightarrow\neg p\equiv\neg q\vee\neg p$ (by the rule $a\rightarrow b\equiv\neg a\vee b$). $p\rightarrow\neg q\equiv\neg p\vee\neg q$. The truth - table for $p\vee q$ is different from the truth - table for $\neg p\vee\neg q$. For $p\vee q$, it is true when either $p$ is true or $q$ is true. For $\neg p\vee\neg q$, it is true when either $p$ is false or $q$ is false.

Answer:

D. None are equivalent