Discrete Mathematics Answers

 Let f be the function from {a, b, c} to {1, 2, 3} such that f(a) = 2, f(b) = 3 and

f(c) = 1. Is f invertible, and if it is, what is it’s inverse?

Construct the truth tables for the following compound propositions.

1. (𝑎 ∧ 𝑏) → (𝑎 ∨ 𝑏)

2. (𝑎 → 𝑏) ∨ (∼ 𝑎 → 𝑏)

3. [(𝑎 → 𝑏) ∧ (𝑎 → 𝑐)] → (𝑎 → (𝑏 ∧ 𝑐))

4. ∼ (𝑏 ∧ 𝑎) ↔ (𝑏 ∨∼ 𝑎)

5. (𝑎 → 𝑏) ∨ 𝑐

Suppose that A = {1,3,5}, B = {1,5}, C = {3,7}, and D = {1,3}. Determine which of these sets are subsets of which other of these sets.

If R⊆S, then T∘R ⊆ T∘S and R∘T ⊆ S∘T

(S∪T)∘R= (S∘R)∪(T∘R).

Draw the directed graph of the relation

R = {(1,1),(1,3),(2,1),(2,3),(2,4),(3,1),(3,2),(4,1)} ,

S = {(1,3),(1,4),(2,1),(2,2),(2,3),(3,1),(3,3),(4,1),(4,3)} ,

Use these graphs to draw the graphs of (a)

𝑅

−

1

&

𝑆

−

1. –p → q
2.  (p → –q) ∨ –q
3.  p → –(p ∨ q)
4. (p ∧ q) ∨ (–p ∨ q)
5. [ (p ∧ q) ∨ –p ] ∧ –q

 Suppose you randomly select k of the first 2016 positive integers. What is the smallest k that guarantees that at least one pair of the selected integers will sum to 2017?

5. You have 5 different-colored bottles, each with a distinct cap. In how many ways can these caps be put on the bottles such that none of the caps are on the correct bottles? (Assume that all the caps must be on the bottles.) 

Given an=an−1−6an−2 where a0=1 a2=5

a.) list the first 10 terms of the sequence

b.) find a closed form(solve the recurrence relations)