Discrete Mathematics Answers

For each of these arguments, explain which rules of inference are used for each

step.

a) “Doug, a student in this class, knows how to write programs in JAVA. Everyone who

knows how to write programs in JAVA can get a high-paying job. Therefore,

someone in this class can get a high-paying job.”

b) “Somebody in this class enjoys whale watching. Every person who enjoys whale

watching cares about ocean pollution. Therefore, there is a person in this class who

cares about ocean pollution.”

c) “Each of the 93 students in this class owns a personal computer. Everyone who

owns a personal computer can use a word processing program. Therefore, Zeke, a

student in this class, can use a word processing program.”

d) “Everyone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey

has never seen the ocean. Therefore, someone who lives within 50 miles of the

ocean has never seen the ocean.”

Let W(x, y) mean that student x has visited website y, where the domain for x

consists of all students in your school and the domain for y consists of all

websites. Express each of these statements by a simple English sentence.

a) W(Sarah Smith, www.att.com)

b) ∃xW(x, www.imdb.org)

c) ∃yW(José Orez, y)

d) ∃y(W(Ashok Puri, y) ∧ W(Cindy Yoon, y))

e) ∃y∀z(y ≠ (David Belcher) ∧ (W(David Belcher, z) → W(y,z))

f) ∃x∃y∀z((x ≠ y) ∧ (W(x, z) ↔ W(y, z)))

Express the negations of these propositions using quantifiers, and in English.

a) Every student in this class likes mathematics.

b) There is a student in this class who has never seen a computer.

c) There is a student in this class who has taken every mathematics course offered at

this school.

d) There is a student in this class who has been in at least one room of every building

on campus

Translate these system specifications into English where the predicate S(x, y) is

“x is in state y” and where the domain for x and y consists of all systems and all

possible states, respectively.

a) ∃xS(x, open)

b) ∀x(S(x, malfunctioning) ∨ S(x, diagnostic))

c) ∃xS(x, open) ∨ ∃xS(x, diagnostic)

d) ∃x¬S(x, available)

e) ∀x¬S(x, working)

Let n be 26. Construct a map of a continent having n different countries in such a way that four colours are needed to colour bordering countries in different colours. Using blue for the ocean (and possibly for some of the countries), and three other colours of your choice for the countries, apply colours to the map. [Use four colours whose names start with different letters, so that you can represent each colour with the first letter of its name.]

(b) By inserting vertices at strategic locations, convert your map into a connected planar graph in which faces separated by an edge are differently coloured.

(c) Construct the dual graph of your connected planar graph, and apply the corresponding vertex colouring to your dual graph

please i need answer in simple form with clear explanation to understand , i need answer as requirement

A. COUNTING METHODS

1. How many strings of length 4 can be formed using the letters ABCDE if it starts with letters AC and repetition is not

allowed?

2. There are 10 multiple choice questions in an examination. Each of the questions have four choices. In how many ways

can an examinee give possible answers?

B. BINOMIAL COEFFICIENTS

Expand (2𝑥 + 4𝑎) 4 using the binomial theorem.

C. PIGEONHOLE PRINCIPLE (5 pts) . Explain briefly.

Do you agree that there are 3 persons who have the same first and last name? Why and why not? 

In a market, 50 women sell only Apple, 25 sell Apple and Banana and 50 sell Banana. Each woman sells at least one the two fruits. How many women are there? 

The relation 'is the father of' is________.

a) reflexive b) irreflexive

c) transitive d) symmetric

Let p: priya is tall and q:priya is beautiful write the following statement in symbolic form.

i) Priya is tall and beautiful

ii) Priya is tall but beautiful

iii) It is false that priya is short or beautiful