1. **State the problem:** We are given propositions: - $p$: I bought a lottery ticket this week. - $q$: I won the million dollar jackpot. We need to express the following logical expressions as English sentences: a) $\neg p$ b) $p \lor q$ c) $p \to q$ d) $p \land q$ e) $p \leftrightarrow q$ f) $\neg p \to \neg q$ g) $\neg p \land \neg q$ h) $\neg p \lor (p \land q)$ 2. **Translate each expression:** a) $\neg p$: "I did not buy a lottery ticket this week." b) $p \lor q$: "I bought a lottery ticket this week or I won the million dollar jackpot (or both)." c) $p \to q$: "If I bought a lottery ticket this week, then I won the million dollar jackpot." d) $p \land q$: "I bought a lottery ticket this week and I won the million dollar jackpot." e) $p \leftrightarrow q$: "I bought a lottery ticket this week if and only if I won the million dollar jackpot." f) $\neg p \to \neg q$: "If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot." g) $\neg p \land \neg q$: "I did not buy a lottery ticket this week and I did not win the million dollar jackpot." h) $\neg p \lor (p \land q)$: "Either I did not buy a lottery ticket this week, or I bought a lottery ticket this week and won the million dollar jackpot."