Let $p$ and $q$ be the propositions
$p :$ I bought a lottery ticket this week.
$q :$ I won the million dollar jackpot.
Express each of these propositions as an English sentence.
a) $\neg p$
b) $p \vee q$
c) $p \rightarrow q$
d) $p \wedge q$
e) $p \leftrightarrow q$
f) $\neg p \rightarrow \neg q$
g) $\neg p \wedge \neg q$
h) $\neg p \vee(p \wedge q)$
