Discrete Mathematics Answers

 Consider the recurrence relation an=4 an−1−4 an−2

Find the general solution to the recurrence relation and the solution when a0=−2 and a1=3.

Find out if the inverse exists for the following, give reasoning behind your answer. If you conclude that the inverse exists then find the B ́ezout coefficients and the inverse of the modulo. [Hint: Example 2 of section 4.4 in the book]

(a) - (3 points) 678 modulo 2970

(b) - (3 points) 137 modulo 2350

Find out if the following numbers are prime numbers, show your work using prime factorization. You may use code to verify your answer but do not put it up as your solution.:

(a) - (3 points) 773

(b) - (3 points) 733

(c) - (3 points) 377

Prove or disprove that there exists exactly two regular graphs

Draw the Venn diagrams for each of these combinations of the sets A, B, and C. Shade

the region(s) corresponding to the given set expressions.

a. (A ∪ B) ∩ C

b. (𝐴̅∩ B) ∩ 𝐶̅

c. (A ∩ 𝐶̅) ∪ 𝐵̅

d. A ∩ (B ∩ 𝐶̅)

Let P = {(1, 4),(3, 5),(4, 1)}, Q = {(1, 5),(2, 2),(3, 4),(5, 2)} and R ={(4, 4),(2, 1),(5, 3),(3, 4)}.

Find P ◦ Q ◦ R

Discussion Assignment

Let f(x)=\sqrt(x) with f: \mathbb{R} \to \mathbb{R}. Discuss the properties of f. Is it injective, surjective, bijective, is it a function? Why or why not? Under what conditions change this?

Explain using examples.

Let 𝑝, 𝑞 and 𝑟 be the propositions:

𝑝: You have the flu

𝑞: You miss the final examination

𝑟: You pass the course

Express each of the following propositions as an English sentence.

i) ¬𝑞 ↔ 𝑟 [2 Marks]

ii) (𝑝 → ¬𝑟)˅(𝑞 → ¬𝑟) [2 Marks]

Show, by the use of the truth table/matrix, that the statement (p v q) v [( ¬p) ∧ (¬q)] is a tautology.

5. What is the negation of each of these propositions? a) Mei has an MP3 player. b) There is no pollution in New Jersey. c) 2 + 1 = 3. d) The summer in Maine is hot and sunny.