**: Let p and q be the propositions. p: “I bought a lottery ticket.” q: “I won the million dollar jackpot on Friday.” Express each of these propositions as an English sentence. a) ￢p b) ￢p →￢q c) p ↔q d) ￢p ∨(p ˄ q)**

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a) The proposition ￢*p is *expressed as "I did not buy a lottery ticket".* *

b) The proposition ￢*p *→￢*q is *expressed as "If I did not buy a lottery ticket, then I did not win the million dollar jackpot on Friday".

c) The proposition *p *↔*q is *expressed as "I bought a lottery ticket if and only if I won the million dollar jackpot on Friday."* *

d) The proposition ￢*p *∨(p ˄ q) is expressed as "I did not buy a lottery ticket or I bought a lottery ticket and I won the million dollar jackpot on Friday".* *