Please help

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Please help

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Consider the sentence (2::P(x) A 3rQ(x)) > Ex(PU) AQ=)). Which of the following is true? A. The sentence is valid. B. The sentence is not valid. It is false under the interpretation where I ranges over the positive integers, P() is '< <3' and Q(1) is : 6 C. The sentence is not valid. It is false under the interpretation where I ranges over the positive integers, P(x)is 'I > 3 and Q() is '6" D. The sentence is not valid. It is false under the interpretation where I ranges over the positive integers, PCC) is' <-3' and Q(x) is ">6 Suppose you know the following about a statement P(n). • P(3), P(5) and P(8) are all true. • P(4) is false. • For all integers k > 6, if P(k) is true then P(k+ 1) is true. What is the smallest integer r for which you can be sure that P(n) is true for all integers n >x? Answer: Which of the following sentences is logically equivalent to -32=y(P(2,y) AQ(x,y))? 5x=y(-P(x,y) A-Q(2,4) 1x y(- P(x,y) V-Q(x,y)) VzVy(-P(x,y) A-Q(x,y)) VaVy(-P(x,y) V-Q(x,y)) Consider the sentences V==yP(1, y) and =yvxP(x,y). Are these sentences true or false under the interpretation where x and y range over the positive integers and Pry) is 'x S 3y"? VrlyP(x,y) is true, P(x,y) is true VrTypx,y) is true, WP(r,y) is false FrüyP(x,y) is false, la P(x, y) is true VrTyPar,y) is false, TaPa y) is false Consider the sentences Vr=yP(x, y) and Dyr P(2,y). Are these sentences true or false under the interpretation where r and y range over the integers and Px,y) is "X = 5y"? Vr=yP(x,y) is true, VrP(x,y) is true PrÐyP(x,y) is true, Wyd.xP(x,y) is false VadyP(x,y) is false, y&P(x,y) is true Vr=yP(x, y) is false, EyVxP(z,y) is false to prove by simple induction that 32 | 12 | 1? > 31% for all integers n>8, the base step would consist of proving which of the following inequalities? 9+16+ 25 + 36 +19+ 61 < (24) 1+4+9+16 + 25 +36 +49 +64 > 192 9 | 16 + 25 | 36 | 49 64 > 192 9+16 + 25 + 36 +19+ 61 > (24) 1+4+9 – 16 + 25 + 36 > 192