1. The problem asks to express logical propositions in English based on the given propositions: - $p$: I bought a lottery ticket this week. - $q$: I won the million dollar jackpot. 2. Part (a): $\neg p$ means "not $p$", so: "I did not buy a lottery ticket this week." 3. Part (b): $p \lor q$ means "$p$ or $q$ or both", so: "I bought a lottery ticket this week or I won the million dollar jackpot." 4. Part (c): $p \to q$ means "if $p$ then $q$", so: "If I bought a lottery ticket this week, then I won the million dollar jackpot." 5. Part (d): $p \land q$ means "$p$ and $q$", so: "I bought a lottery ticket this week and I won the million dollar jackpot." 6. Part (e): $p \leftrightarrow q$ means "$p$ if and only if $q$", so: "I bought a lottery ticket this week if and only if I won the million dollar jackpot." 7. Part (f): $\neg p \to \neg q$ means "if not $p$ then not $q$", so: "If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot." 8. Part (g): $\neg p \land \neg q$ means "not $p$ and not $q$", so: "I did not buy a lottery ticket this week and I did not win the million dollar jackpot." 9. Part (h): $\neg p \lor (p \land q)$ means "not $p$ or both $p$ and $q$", so: "Either I did not buy a lottery ticket this week, or I bought one and won the million dollar jackpot." Final answers are the English sentences for each part.