Discrete Mathematics Answers

For the following relation on the set {x:x∈Ζ and 1 ≤ x ≤ 12}. List 

(a) the ordered pairs belonging to the relation:
(b) the transitive closure

R = {(x, y): xy = 9}

Find the number of integers between 1 and 300 both inclusive that are divisible by the

following types:

(i). at-least one of 3, 5, 7.

(ii). 3 and 5 but not 7.

(iii). 5 but neither 3 nor 7

Identify if the following statements are predicate logic. Give a domain of discourse for each propositional function.

A. The movie won the Academy Award for the past 2 years.

B. 1 + 3 = 4

C. (x+2)2 is a prime number.

In a class of 32 students, 18 offers chemistry, 16 offers Physics 22 offer Mathematics. 6 offer all three subjects, 3 offer Chemistry and Physics only and 5 offer only physics only. Each student offers at least one subject. Find the number of student who offer A. chemistry only B. only one student C. only two subjects

Let n be an integer. Use Definition 1.6 to explain why 2n + 9

 is an odd integer 2n + 9 = 2(    )+ 1

Define a relation  on the set S of all strings of letters: two strings are related if you can get one from the other by reversing one pair of adjacent letters. For example, cow  ocw

 but cow  woc.

Q1. a) Let 𝑈 = {𝑥: 𝑥 ∈ 𝑍, 1 ≤ 𝑥 ≤ 12},

𝐴 = {2𝑥: 𝑥 ∈ 𝑈 𝑎𝑛𝑑 𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 4},

𝐵 = {𝑥: 𝑥 ∈ 𝑈, 𝑥 𝑑𝑖𝑣𝑖𝑑𝑒𝑠 2} 𝑎𝑛𝑑 𝐶 = {𝑥: 𝑥 ∈ 𝑈, 𝑥2 ≤ 16}.

i) List the elements belong to the sets A, B and C respectively. (3 marks)

ii) Find 𝐶 − (𝐴̅ ∩ 𝐵) (3 marks)

iii) Find 𝐵⊕𝐶̅. (2 marks)

b) Prove by induction that 1 + 5 + 9 + … + (4n – 3) = n(2n – 1) for all n ≥ 1. (5 marks)

c) Let 𝑥 = 866 𝑎𝑛𝑑 𝑦 = 732.

(i) Find the greatest common divisor of x and y and then express it in the form of

ax + by, where 𝑎, 𝑏 ∈ 𝑍. (5 marks)

(ii) Find the least common multiple of x and y. (2 marks)

[Total: 20 marks]

Prove that 2/n⁴-3 if and only off 4/n²+3

Determine the validity of the following argument:

Having a strong mathematical background is necessary and sufficient

for understanding the concepts in Information Technology. Having a 

strong mathematical background and understanding the concepts in 

Information Technology is not sufficient for you to pass all Information 

Technology exams. Therefore, if you do not pass all Information 

Technology exams it is not the case that either you did not have a strong 

mathematical background or you understood the concepts in 

information Technology.