1.(a) If the tuple (a, b, c) satisfy the following conditions:
• a, b and c are consecutive odd integers.
• a, b and c are all primes.
Then we call it a super prime tuple. Prove that (3, 5, 7) is the only super prime tuple.
(b) Define F0 = 0, F1 = 1, Fn+2 = Fn+1 + Fn for n = 0, 1, 2, .... Prove that for any n ≥ 0 we have
Fn ≤ ((1 + 5 )/2)^n - 1.
2.
(a) Assume that ∀x∃yP(x, y) is false and the domain of them is nonempty. Which of the following must be false?
(i) ∀x∀yP(x, y)
(ii) ∃x∀yP(x, y)
(iii) ∃x∃yP(x, y)
(b) Let P(x) denote the statement "x is an accountant" and let Q(x) denote the statement "x owns a Porsche". Write each statement below in first order logic.
(i) All accountants own Porsches.
(ii) Some accountant owns a Porsche.
(iii) All owners of Porsches are accountants
(iv) Someone who owns a Porsche is an accountant
3.
(a) Prove ⌈x⌉ = - ⌊- x⌋ for any real number x
(b) A detective has interviewed four witnesses to a crime. From their stories, the detective has concluded that:
(i) If the butler is telling the truth, then so is the cook.
(ii) The cook and the gardener cannot both be telling the truth.
(iii) The gardener and the handyman are not both lying.
(iv) If the handyman is telling the truth then the cook is lying.
(v) Who must be lying? There may be more than one liar. Show your steps
4. Define the sequence c0, c1, ... by the equations c0 = 0 and cn = c⌊n/2⌋ + 3 for all n > 0. Prove that cn ≤2n for all n ≥ 3.
5.
(a) There are k kinds of postcards, each with a limited amount. Let there be ai copies of the i-th postcard. How many ways are there to send these postcards to n friends? (Each friend can receive zero cards, as well as multiple copies of the same card.)