Discrete Mathematics Answers

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

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?

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

Type of relations for aRb if and only if a<= b+1

Solve the following. (10 pts each)

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

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

From around 1 pm to 5 pm, you attended the party of your friend. Because you are close to each other, she let you get her gold necklace in the jewelry box on the table in her room. You agreed to your friend’s request to get the necklace in her room. As you enter the room and open the jewelry box, you see nothing. Your friend calls the police to accuse you that you steal her gold necklace. How can you prove that you are innocent? Give your statements. 

Find the edge chromatic

numbers to colour the edges of the complete graph with four and five vertices

Let f be a function from A to B. Find f^-1

let P(x) denote the statement "x-3>5" . What are the truth values? P(2) P(8) P(9)

Generate the output in each set:

(a)    The set of all positive even numbers less than or equal to 10. (2 marks)

(b)    The set of all letters in the word "PURAKENCANA". (2 marks)

(c)    The set of all whole numbers greater than 3 and smaller than 16, and divisible by 3. (2 marks)

(d)    The set of all prime numbers divisible by 3. (2 marks)

(e)    The set of all numbers whose absolute value is equal to 7. (2 marks)

Let p and q be the propositions;

p: You drive over 110 kilometre per hour.

q: You get a speeding summons.

Write the following propositions using p and q and logical connectives.

a. You drive over 110 kilometre per hour, but you do not get a speeding summons.

(2 marks)

b. You get a speeding summon, but you do not drive over 110 kilometre per hour.

(2 marks)

c. If you do not drive over 110 kilometre per hour, then you will not get a speeding summon.

(2 marks)

d. Driving over 110 kilometre per hour is sufficient for getting a speeding summon.

(2 Marks)