Give asymptotically tight upper and lower bounds for T (n) in each of the following algorithmic recurrences. Justify your answers. E. ?(?) = 3?((?/3) - 2) + ?/2 (Hint: think about how you can use an assumption about the importance of the -2 to apply the Masters Theorem)

f(n) = n3 - (n2 log2 n) + 2n f(n) ∈ O(n3) by giving the constants (c, n0)  and arguing that your constants hold as n goes to infinity by either "chaining up", "chaining down", carefully treating the inequality as an equality and doing some algebra, or even weak induction. Also, what is the minimum value for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.

Prove ((A --> B) C) is a proposition.

6. Use rules of inference to show that if Vr(P(x) v Q(x)) and Vr((-P(x) ^ Q(x)) → R(x)) are true, then Vx(-R(x) → P(x)) is also true, where the domains of all quantifiers are the same. 7. (a) Prove that m³ + 2n² = 36 has no solution in positive integers. (b) Prove that for every n e Z, n³ +n is even. (c) Prove that for all n e Z, n is odd if and only if n + 2 is odd. (d) Prove that the product of two consecutive integers is even.