1. **Problem statement:** Translate the English statements into propositional logic expressions with clear proposition definitions. 2. **Define propositions for (a):** - $S$: The server is running. - $D$: The database is corrupted. - $B$: The backup system will activate. - $L$: Data will be lost. - $A$: The administrator intervenes immediately. 3. **Translate (a):** - "The server is not running or the database is corrupted" translates to $\neg S \lor D$. - "Either the backup system will activate or data will be lost" translates to $B \lor L$. - "Unless the administrator intervenes immediately" means if $A$ then the previous condition may not hold, so the implication is modified. 4. **Full expression for (a):** $$ (\neg S \lor D) \to ((B \lor L) \land \neg A) $$ 5. **Define propositions for (b):** - $P$: The student passes the course. - $F$: The student scores above 85% on the final exam. - $A$: The student has perfect attendance. - $S$: The student scores above 70%. - $G$: The student plagiarized. 6. **Translate (b):** - "Passes if and only if" means $P \leftrightarrow$ (condition). - Condition: either $F$ or ($A$ and $S$), provided they did not plagiarize ($\neg G$). 7. **Full expression for (b):** $$ P \leftrightarrow ((F \lor (A \land S)) \land \neg G) $$ **Final answers:** (a) $$ (\neg S \lor D) \to ((B \lor L) \land \neg A) $$ (b) $$ P \leftrightarrow ((F \lor (A \land S)) \land \neg G) $$