Q1: Prove that n2-7n 12 is non negative whenever n is integer and n 2 3 (4pt) Q2

Question

Q1: Prove that n2-7n 12 is non negative whenever n is integer and n 2 3 (4pt) Q2: Given: p: I bought a lottery ticket this week. q: I won the million dollar jackpot on Friday Express each of these propositions as English sentences (3pt) a)pq 03: Construct a truth table for each of these formulas: (3pt) Q4: Determine whether [(pVqJA(p r)A(q r)] r is a tautology (4pt) Q5: Determine whether [-pNp q)--q is a tautology (4pt) Q6: Prove that [(-rVf) (sADJA[s-t]At1-r is theorem using deductive proof method (4pt) Q7: Let F(x,y) = "X can fool Y". Domain-"all people in the world". Use quantifiers to express each of these statements (3pt) a) Everybody can fool Fred b) Everybody can fool somebody c) There is no one who can fool everybody

Explanation / Answer

AS PER CHEGG'S GUIDELINES, ANSWERING THE FIRST 4 QUESTIONS ONLY:

Q1:

Basis step: Let n = 3. Then

n2 7n + 12 = 32 7 · 3 + 12 = 9 21 + 12 = 0. I

nductive hypothesis: Assume for some integer k 3 that k2 7k + 12 is nonnegative.

Inductive step: (k + 1)2 7(k + 1) + 12 = k2 + 2k + 1 7k 7 + 12

= (k2 7k + 12) + (2k + 1 7)

0 + 2k + 1 7

= 2k 6 2 · 3 6

= 0

Q2:

a. p -> q

If I buy a ticket, I will win the million dollar jackpot

b. ~p -> ~q

  If I don't buy a lottery ticket this week then I will not win the million dollar jackpot

c. ~p ^ ~q

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

Q3:

a)

b)

c)

Q4:

Since [(p q) (p r) (q r)] r is always T, it is a tautology.

p q p ¬q F F F F T T T F T T T F

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