Say whether the following are correct. If yes, provide a natural deduction proof. If not, provide a counterexample.  P → Q ` (R → P) → (R → Q)

Subject: algebra

Prove 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 val4ue for c (in the definition of O() ) forj this problem and is it an exclusive or inclusive minimum/bound.

Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not logically equivalent, without using truth tables.

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.

Show that (p ∧ q) → r and (p → r) ∧ (q → r) are not logically equivalent, without using truth tables.

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

course (discrete structure) help plz

¬P ∧¬Q≡¬(P ∨Q)   Prove the following equivalency relation (Proof, Justification/explanation):

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