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Discrete structure

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Tutorial 3
4 [8] Translate these statements into English, where
R(x)
is “
x
is a rabbit” and
H(x)
is “
x
hops” and the
domain consists of all animals.
a)
∀
x(R(x)
→
H(x))
b)
∀
x(R(x)
∧
H(x))
c)
∃
x(R(x)
→
H(x))
d)
∃
x(R(x)
∧
H(x))
6 [10] Let
C(x)
be the
statement “
x
has a cat,” let
D(x)
be the statement “
x
has a dog,” and let
F(x)
be the
statement “
x
has a ferret.” Express each of these statements in terms of
C(x)
,
D(x)
,
F(x)
, quantifiers, and logical connectives. Let the domain consist of all students in your class. a)
A student in your class has a cat, a dog, and a ferret. b)
All students in your class have a cat, a dog, or a ferret. c)
Some student in your class has a cat and a ferret, but not a dog. d)
No student in your class has a cat, a dog, and a ferret. e)
For each of the three animals, cats, dogs, and ferrets, there is a student in your class who has this animal as a pet. 8 [11] Let
P(x)
be the statement “
x
=
x
2.” If the domain consists of the integers, what are these truth values?
a)
P(
0
)
b)
P(
1
)
c)
P(
2
)
d)
P(
−
1
)
e)
∃
xP(x)
f )
∀
xP(x)
12 [19] Suppose that the domain of he propositional function
P(x)
consists of the integers 1, 2, 3, 4, and 5. Express these statements without using quantifiers, instead using only negations, disjunctions, and conjunctions.
a)
∃
xP(x)
b)
∀
xP(x)
c)
￢
∃
xP(x)
d)
￢
∀
xP(x)
e)
∀
x((x
_= 3
)
→
P(x))
∨
∃
x
￢
P(x)
14[24] Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people.
a)
Everyone in your class has a cellular phone.
b)
Somebody in your class has seen a foreign movie.
c)
There is a person in your class who cannot swim.
d)
All students in your class can solve quadratic equations.
e)
Some student in your class does not want to be rich.
16[26] Translate each of these statements into logical expressions in three different ways by varying the domain and by using predicates with one and with two variables.
a)
Someone in your school has visited Uzbekistan.
b)
Everyone in your class has studied calculus and C++.
c)
No one in your school owns both a bicycle and a motorcycle.
d)
There is a person in your school who is not happy.
e)
Everyone in your school was born in the twentieth century. 19[31] Suppose that the domain of
Q(x, y, z)
consists of triples
x, y, z
, where
x
= 0
,
1
,
or 2,
y
= 0 or 1, and
z
= 0 or 1. Write out these propositions using disjunctions and conjunctions.
a)
∀
yQ(
0
, y,
0
)
b)
∃
xQ(x,
1
,
1
)
c)
∃
z
￢
Q(
0
,
0
, z)
d)
∃
x
￢
Q(x,
0
,
1
)
24[40] Express each of these system specifications using predicates, quantifiers, and logical connectives.
a)
When there is less than 30 megabytes free on the hard disk, a warning message is sent to all users.
b)
No directories in the file system can be opened and no files can be closed when system errors have been detected.
c)
The file system cannot be backed up if there is a user currently logged on.
d)
Video on demand can be delivered when there are at least 8 megabytes of memory available and the connection speed is at least 56 kilobits per second. 4 [7] Let
T (x, y)
mean that student
x
likes cuisine
y
, where the domain for
x
consists of all students at your school and the domain for
y
consists of all cuisines. Express each of these statements by a simple English sentence.
a)
￢
T
(Abdallah Hussein, Japanese)
b)
∃
xT (x
, Korean)
∧
∀
xT (x,
Mexican)
c)
∃
y(T
(Monique Arsenault,
y)
∨
T
(Jay Johnson,
y
))
d)
∀
x
∀
z
∃
y((x
_=
z)
→
￢
(T (x, y)
∧
T (z, y)))
e)
∃
x
∃
z
∀
y(T (x, y)
↔
T (z,y))
f)
f )
∀
x
∀
z
∃
y(T (x, y)
↔
T (z,y))
6 [8] Let
Q(x, y)
be the statement “student
x
has been a contestant on quiz show
y
.” Express each of these
sentences in terms of
Q(x, y)
, quantifiers, and logical connectives, where the domain for
x
consists of all students at your school and for
y
consists of all quiz shows on television.
a)
There is a student at your school who has been a contestant on a television quiz show.
b)
No student at your school has ever been a contestant on a television quiz show.
c)
There is a student at your school who has been a contestant on
Jeopardy
and on
Wheel of Fortune
.
d)
Every television quiz show has had a student from your school as a contestant.
e)
At least two students from your school have been contestants on
Jeopardy
.
10 [19] Express each of these statements using mathematical and logical operators, predicates, and quantifiers, where the domain consists of all integers.
a)
The sum of two negative integers is negative.
b)
The difference of two positive integers is not necessarily positive.
c)
The sum of the squares of two integers is greater than or equal to the square of their sum.
d)
The absolute value of the product of two integers is the product of their absolute values. 16 [30] Rewrite each of these statements so that negations appear only within predicates (that is, so that no negation is outside a uantifier or an expression involving logical connectives).
a)
￢
∃
y
∃
xP(x, y)
b)
￢
∀
x
∃
yP(x, y)
c)
￢
∃
y(Q(y)
∧
∀
x
￢
R(x, y))
d)
￢
∃
y(
∃
xR(x, y)
∨
∀
xS(x, y))
e)
￢
∃
y(
∀
x
∃
zT (x, y, z)
∨
∃
x
∀
zU(x, y, z))
20 [38] 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.
Tutorial 4
2 [3] What rule of inference is used in each of these arguments?
a)
Alice is a mathematics major. Therefore, Alice is either a mathematics major or a computer science major.
b)
Jerry is a mathematics major and a computer science major. Therefore, Jerry is a mathematics major.
c)
If it is rainy, then the pool will be closed. It is rainy. Therefore, the pool is closed.
d)
If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today.
e)
If I go swimming, then I will stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn.
4 [6] Use rules of inference to show that the hypotheses “If it does not rain or if it is not foggy, then the
sailing race will be hel
d and the lifesaving demonstration will go on,” “If the sailing race is held, then the trophy will be awarded,” and “The trophy was not awarded” imply the conclusion “It rained.”
6 [8] What rules of inference are used in this argument? “No man is an isla
nd. Manhattan is an island.
Therefore, Manhattan is not a man.”
8[14] For each of these arguments, explain which rules of inference are used for each step. a)
“Linda, a student in this class, owns a red convertible. Everyone who owns a red convertible has gotten at least one speeding ticket. Therefore, someone in this class has gotten a speeding ticket.”
b)
“Each of five roommates, Melissa, Aaron, Ralph,Veneesha, and Keeshawn, has taken a course in
discrete mathematics. Every student who has taken a course in discrete mathematics can take a
course in algorithms. Therefore, all five roommates can take a course in algorithms next year.”
c)
“All movies produced by John Sayles are wonderful. John Sayles produced a movie about coal
miners. Therefore, there is a wonderf
ul movie about coal miners.”
d)
“There is someone in this class who has been to France. Everyone who goes to France visits the
Louvre. Therefore, someone in this class has visited
the Louvre.”
16[24] Identify the error or errors in this argument that supposedly shows that if
∀
x(P(x)
∨
Q(x))
is true then
∀
xP(x)
∨
∀
xQ(x)
is true. a)
∀
x(P(x)
∨
Q(x))
Premise b)
P(c)
∨
Q(c)
Universal instantiation from (1) c)
P(c)
Simplification from (2) d)
∀
xP(x)
Universal generalization from (3) e)
Q(c)
Simplification from (2) f)
∀
xQ(x)
Universal generalization from (5) g)
∀
x(P(x)
∨
∀
xQ(x))
Conjunction from (4) and (6)

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