Question

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)$p\to q$

d) $p\wedge q$

e)$p\leftrightarrow q$

f )$\neg p\to \neg q$

g)$\neg p\wedge \neg q$

h)$\neg p\vee (p\wedge q)$

Short answer

1. I did not buy any lottery tickets this week.
2. I bought a lottery ticket or won the million-dollar jackpot.
3. If I bought a lottery ticket this week then I would have won the million-dollar jackpot.
4. I bought a lottery ticket this week and I won the million-dollar jackpot.
5. I bought a lottery ticket this week if and only if I won the million-dollar jackpot.
6. If I did not buy a lottery ticket this week then I did not win the million-dollar jackpot.
7. I did not buy a lottery ticket this week and I did not win the million-dollar jackpot.
8. I did not buy a lottery ticket this week or I bought a lottery ticket this week and I won the million-dollar jackpot.

Step-by-step solution

Definition of the proposition

Proposition:
1. It is a declarative statement that can be either true or false.
2. It cannot be both true and false simultaneously

English sentence for a)

p is I bought a lottery ticket this week.

$\neg p$would mean I did not buy a lottery ticket this week.

English sentence for b)

p is I bought a lottery ticket this week.
q is I won the million-dollar jackpot

$p\vee q$means,

I bought a lottery ticket this week or I won the million dollar jackpot.

English sentence for c)

We write given statement as,

$p\to q$, if p then q means,

If I bought a lottery ticket this week then I would have won the million dollar jackpot.

English sentence in d)

We write given statement as,

$p\wedge q$,p and q means,

I bought a lottery ticket this week and I won the million dollar jackpot.

English sentence in e)

We write given statement as,

$p\leftrightarrow q$,p iff q means,

I bought a lottery ticket this week if and only if I won the million dollar jackpot.

English sentence in f)

We write given statement as,

$\neg p\to \neg q$,If not p then not q means,

If I did not buy a lottery ticket this week then I did not win the million dollar jackpot.

English sentence in g)

We write given statement as,

$\neg p\wedge \neg q$, not p and not q means,

I did not buy a lottery ticket this week and I did not win the million dollar jackpot.

English sentence in h)

We write the given statement as,

$\neg p\vee (p\wedge q)$, not p or p and q means,

I did not buy a lottery ticket this week or I bought a lottery ticket this week and I won the million-dollar jackpot.