Discrete Mathematics Answers

5.     Suppose that G is a connected multigraph with 2k vertices of odd degree. Show that there exist k subgraphs that have G as their union, where each of these subgraphs has a Euler path and where no two of these subgraphs have an edge in common.

Determine whether each of the following is true or false.

a. 0 ∈ ø

b. ø ∈ {0}

c. {ø} ⊆ {0}

d. {ø} ⊆ {ø}

e. ø ∈ {0, ø}

In how many different ways can you put20 balls of the same colour into two numbered box

Let 𝑃𝑃(𝑛𝑛) be the proposition that 1(1!) + 2(2!) + 3(3!) + ⋯+ 𝑛𝑛(𝑛𝑛!) = (𝑛𝑛+ 1)! −1. Prove by induction that 𝑃𝑃(𝑛𝑛) is true for all 𝑛𝑛≥1.

Prove that the product of any three consecutive integers is a multiple of 3. 

Use a proof by contraposition to show that if 𝑛𝑛2 + 1 is even, then 𝑛𝑛 is odd.

Use a direct proof to show that every odd integer is the difference of two squares.  

2. Use generating functions to solve the recurrence relation a_n = 4a_n−1 − 4a_n−2 + n², where a₀ = 2, a₁ = 5.

List the quadruples in the relation { a,b,c,d} where a, b, c, d are integers with

0 < a < b < c < d < 8.

Let R be the relation on the set {0, 1, 2, 3} containing the ordered pairs (0,1),(1, 1), (1, 2), (2, 0), (2, 2) and (3, 0). Find the

(i) reflexive closure of R, (ii) symmetric closure of R