Discrete Mathematics Answers

1.   For the following functions, provide the domain and range

a)  f: . For each x ∈ domain f(x) = cx.

b)  g:  . For each x,y ∈ domain, g(x,y) = yxd.

Draw a simple, undirected graph yourself, the vertices are connected with each other including 8 vertices and 14 edges. Find the shortest path from two arbitrary vertices:​

a) The weight of each edge is 1.​

b) Self-weighting for edges

 Show that p ↔ q and (p ∧ q) ∨ (¬p ∧ ¬q) are logically equivalent.

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