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

Equation Transcription:

Text Transcription:

p

q

Neg p

P vee q

p rightarrow q

p wedge q

p leftrightarrow q

p rightarrow q

Nega p rightarrow neg q

Nega p wedge neg q

Nega p vee (p wedge q)

Solution:

Step1

Given that

I bought a lottery ticket this week and I won the million dollar jackpot be a preposition p and q.

Step2

To find

Express each of these propositions as an English sentence.

Step3

From the given proposition we are finding that

a.

This means this week i have not buy a lottery ticket.

Step4

b.

This means this week either i bought a lottery ticket or i won the million dollar jackpot.

Step5

c.

This means this week if i bought a lottery ticket then the million dollar jackpot i won.

Step6

d.

This means this week if i bought a lottery ticket and the million dollar jackpot i won.

Step7

e.

This means this week if i bought a lottery ticket if and only if the million dollar jackpot i won.

Step8

f.

This means this week if i have not bought a lottery ticket then the million dollar jackpot i have not won.

Step9

g.

This means this week if i have not bought a lottery ticket and the million dollar jackpot i have not won.

Step10

h.

This means either this week i have not bought a lottery ticket or else i have and the million dollar jackpot i won.