Discrete Mathematics Answers

Ac - Cc

x is a member of A

U= {1, 2, 3, 4, 5, 6, 7, 8, }, A= {1, 2, 3, 4, 5, 7}, B= {1, 5, 6, 7}, C= {1, 2, 3, 6}

16-20. Illustrate the Venn Diagram for the sets A, B, and C.

10. 𝐴 ⊂ 𝐵

Suppose T(n) and f(n) and two functions. Write asymptotic notations (Ο, Ω, Θ) using these two functions and explain the growth rate of these functions in each notation.

Find a recurrence relation, with initial condition, that uniquely determines each of the following geometric progressions.

a)        2, 10, 50, 250, . . .

b)        7, 14/5, 28/25, 56/125, . . .

Prove that 2 − 2 · 7 + 2 · 7² −· · ·+2(−7)ⁿ = (1 – ((−7)ⁿ⁺¹)/4 whenever n is a nonnegative integer

Let D = {-5, -3, -1, 1, 3, 5}. Write the following statements using only negations, conjunctions and

disjunctions:

a) ∃𝑥𝑃(𝑥)

b) ∀𝑥𝑃(𝑥)

c) ∀𝑥((𝑥 ≠ 1) → 𝑃(𝑥))

d) ∃𝑥((𝑥 ≥ 0) ∧ 𝑃(𝑥))

e) ∃𝑥(￢𝑃(𝑥)) ∧ ∀𝑥((𝑥 < 0) → 𝑃(𝑥))

Write the negation of the following statement:

∀𝑥∃𝑦(𝑥 + 𝑦 = 2 ∧ 2𝑥 − 𝑦 = 1

Prove by mathematical induction that n^2 +n < 2^n whenever n is an integer greater than 4.