1. Problem statement: We need to find the symbolic compound statement for "If she will not pass the exams, then she is sad if and only if she is not good in math or she is not doing well in English." 2. Define variables: - $p$: She is not good in math. - $q$: She is doing well in English. - $r$: She will pass the exams. - $s$: She is happy. 3. Analyze the sentence: - "If she will not pass the exams" means $\neg r$. - "then she is sad" means the consequent is $\neg s$ (because $s$ is happy, so sad is $\neg s$). - "if and only if" corresponds to biconditional $\leftrightarrow$. - "she is not good in math or she is not doing well in English" translates to $p \lor \neg q$ (since $p$ is "not good in math", and $q$ is "doing well in English", its negation is "not doing well in English"). 4. Construct the statement step-by-step: - "If she will not pass the exams, then she is sad" is $\neg r \to \neg s$. - The entire sentence says this implication is equivalent to ("if and only if") $p \lor \neg q$. 5. Final symbolic statement: $$ (\neg r \to \neg s) \leftrightarrow (p \lor \neg q) $$ 6. Compare to answer options: - Option a: $(\neg r \to \neg s) \leftrightarrow (p \lor q)$ (incorrect because $q$ is not negated) - Option b: $(r \to \neg s) \leftrightarrow (p \lor q)$ (incorrect because $r$ is not negated) - Option c: $(\neg r \to s) \leftrightarrow (p \lor \neg q)$ (incorrect because consequent is $s$, not $\neg s$) - Option d: $(\neg r \to \neg s) \leftrightarrow (p \lor \neg q)$ (correct) Answer: d