Discrete Mathematics Answers

Determine whether the function f is a bijection from R to R ? Find fog and gof where f(x)=2x²+3 and g(x)=x=1

A. COUNTING METHODS (5 pts each)

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. (10 pts)

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? 

The set of all natural number whose square is more than 21. 

A = Q, (a*b) = a + b

What is the Cartesian product A × B × C, where A is the set of all airlines and B and C are both set of allcities in USA? Give an example of how this Cartesian product can be used

Identify the error or errors in this argument that supposedly shows that if ∃xP

(x) ∧ ∃xQ(x) is true then ∃x(P (x) ∧ Q(x)) is true.

a) ∃xP (x) ∨ ∃xQ(x) Premise

b) ∃xP (x) Simplification from (1)

c) P (c) Existential instantiation from (2)

d) ∃xQ(x) Simplification from (1)

e) Q(c) Existential instantiation from (4)

f) P (c) ∧ Q(c) Conjunction from (3) and (5)

g) ∃x(P (x) ∧ Q(x)) Existential generalization

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)