Discrete Mathematics Answers

As you now that

0^{3 0}

1^{3 1}

2^{3 8}

3^{3 27}

4^{3 64}

5^{3 125}

so, we have a proof that every positive integer is the sum of the cubes of eight non negative integers. For example

7= 1³ + 1³ + 1³ +1³+1³+1³+ 1³ +0³

12= 2³+ 1³ + 1³ + 1³ +1³+0³+0³+0³

Can you disprove this statement?

Show that if x is an integer then x2+x-41= 0 produce prime numbers. 

Proof by contradiction that if n is a positive integer, then n is odd if and only 

if 5n + 6 is odd

 Suppose you have a graph with v vertices and e edges that satisfies v=e+1. Must the graph be a tree? Prove your answer.

p → ~q

MATHEMATICAL INDUCTION AND RECURRENCE 

5. If P(k) = k2 (k + 2)(k – 1) is true, then what is P (k + 1)? (2 pts)

A. (k + 1)2 (k + 2)(k)

B. (k + 1)2 (k + 2)(k)

C. (k + 1)(k + 3)(k)

D. (k + 1)2 (k + 3)(k)

6. Using the principle of mathematical induction, 2n-1 is divisible by which of

the following? (2 pts)

A. 1

B. 0

C. 4

D. ½ 

7. A relation represents an equation where the next term is dependent on the

previous term is called

A. Binomial relation

B. Recurrence relation

C. Regression relation

D. None of these

8. Calculate the value of a2 for the recurrence relation an=17an-1+30n, where

a0=3. (2 pts)

A. 2346

B. 1296

C. 1437

D. 5484 

9. The recurrence relation for Fibonacci sequence is

A. Fn = Fn + 1 + Fn - 2

B. Fn = Fn - 1 + Fn - 2

C. Fn = Fn - 1 - Fn - 2

D. None of these

10. In recurrence relation, a0 represents

A. Current value

B. Starting value

C. The value of next term in the sequence

D. None of these

 MATHEMATICAL INDUCTION AND RECURRENCE

1. What is the base case for inequality 3n > n2 , where n = 2? (2 pts)

A. 3 > 1

B. 9 > 4

C. 6 > 4

D. 4 < 9 

2. For the mathematical induction to be true, what type of number should be

the value of n?

A. natural number

B. imaginary number

C. rational number

D. whole number

3. What would be the hypothesis of the mathematical induction for x(x + 1) <

x! , where x ≥ 7?

A. It is assumed that at x = k, k(k + 1)! < k!

B. It is assumed that at x = k, k(k + 1)! > k!

C. It is assumed that at x = k, k(k + 1)! < (k + 1)!

D. It is assumed that at x = k, k(k + 1)(k + 2)! < k! 

4. For any positive integer x, ________ is divisible by 5 (2 pts)

A. 5x2 + 5

B. 2x + 4

C. x4 + 5x

D. 3x2 + 2

MATHEMATICAL INDUCTION AND RECURRENCE

Solve the following. (10 pts each)

1. Prove P(n) = n2 (n + 1)

2. Recurrence relation an = 2n with the initial term a1 = 2.

RULE OF INFERENCE. Determine if the following argument is valid. If it is valid, what rule of inference is

used in each of the following arguments? Show solution. (4 pts each)

1. Joy wrote a C++ source code, or Jen wrote a Java source code. If Joy wrote a C++ source code, then the

problem was solved. If Jen wrote a Java source code, then the problem was solved.

2. There does not exist someone who likes to be COVID – 19 positive; hence, everyone does not like to be

vaccinated. 

C. RELATION. Consider X = {-3, -2, -1, 0, 1} defined by (x,y) ∈ R if x≥ y.

Find for:

1. Elements of R (3 pts)

2. Domain and Range of R (2 pts)

3. Draw the digraph (3 pts)

4. Identify the properties of R (2pts)