Chapter 2 logics (cont ) discrete structures for computing on august 31, 2017

Discrete Structures for Computing on August 31, 2017 Nguyen An Khuong, Tran Tuan Anh, Le HongTrang Faculty of Computer Science andEngineering University of Technology - VNUHCM Nguyen An

Chapter 2

Logics (cont.)

Discrete Structures for Computing on August 31, 2017

Nguyen An Khuong, Tran Tuan Anh, Le HongTrang Faculty of Computer Science andEngineering University of Technology - VNUHCM

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

1 Predicate Logic

2 Exercise

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

Contents Predicate Logic Exercise

Limits of Propositional Logic

x > 3

All square numbers are not prime numbers 100 is a

square number Therefore 100 is not a prime number

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic Exercise

Definition

variables If values are assigned to all the variables in a

Contents Predicate Logic Exercise

Contents Predicate Logic Exercise

Truth value

x > 3 is true or false?

5 > 3

For every number x; x > 3 holds

There is a number x such that x > 3

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

8xP (x) = P (x) is T for all x

9xP (x) = There exists an element x such that P (x) is T

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Let P (x) be the statement x < 2 What is the truth value of the

quantification 8xP (x), where the domain consists of all real

What is the truth value of the quantification 9xP (x), where

the domain consists of all real number?

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Express the statement Some student in this class comes

from Central Vietnam

Solution 1

M(x) = x comes from Central Vietnam

Domain for x is the students in the class

Contents Predicate Logic

Exercise

Negation of Quantifiers

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Exam pleAll

CSE

tudents stud

y Discrete Math1C(x) deno

te x

is a CSE student

Let S(x) denote x studies Discrete Math 1

8x : C(x) ! S(x)

9x : :(C(x) ! S(x)) 9x : C(x) ^ :S(x)

There is a CSE student who does not study

Discrete Math 1 ContentsPredicate Logic

Exercise

2.11

Another Example

Example

Translate these:

All lions are fierce

Some lions do not drink coffee

Some fierce creatures do not drink coffee

Solution

Let P (x), Q(x) and R(x) be the statements x is a lion , x is

fierce and x drinks coffee , respectively

Contents

Predicate Logic

Exercise

The Order of Quantifiers

The order of quantifiers is important, unless all the quantifiers

are universal quantifiers or all are existential quantifiers

Read from left to right, apply from inner to outer

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Predicate Logic

Exercise

Translating Nested Quantifiers

Example

8x (C(x) _ 9y (C(y) ^ F (x; y)) )

Provided that:

C(x): x has a computer,

F (x; y): x and y are friends,

x; y 2 all students in your school

Answer

For every student x in your school, x has a computer or there is a

student y such that y has a computer and x and y are friends

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

Translating Nested Quantifiers

Example

9x8y8z (((F (x; y) ^ F (x; z) ^ (y 6= z)) ! :F (y; z)))

Provided that:

F (x; y): x; y are friends

x; y; z 2 all students in your school

Answer

There is a student x, so that for every student y, every student

z not the same as y, if x and y are friends, and x and z are

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

Translating into Logical Expressions

Example

1 There is a student in the class has visited Hanoi

2 Every students in the class have visited Nha Trang or

Answer

Assume:

C(x) : x has visited Hanoi

D(x) : x has visited Nha Trang

E(x) : x has visited Vung Tau

Contents Predicate Logic Exercise

Translating into Logical Expressions

Contents Predicate Logic Exercise

Translating into Logical Expressions

Example

If a person is a woman and a parent, then this person is

mother of some one

Contents

Predicate Logic

Exercise

Example

If I have a girlfriend, I will take her to go shopping

Whenever I and my girlfriend go shopping and that day is a

special day, I will surely buy her some expensive gift

If I buy my girlfriend expensive gifts, I will eat noodles

for a week

Today is March 8

March 8 is such a special day

Therefore, if I have a girlfriend,

I will eat noodles for a week

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

Propositional Rules of Inferences

Rule of Inference

p

p ! q) q

:q

p ! q) :p

p ! q

q ! r

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Propositional Rules of Inferences

Rule of Inference

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

If it rains today, then we will not have a barbecue today If we

do not have a barbecue today, then we will have a barbecue

tomorrow Therefore, if it rains today, then we will have a

barbecue tomorrow

Solution

p: It is raining today

q: We will not have a barbecue today

r: We will have barbecue tomorrow

Contents Predicate Logic Exercise

It is not sunny this afternoon

(:p) and it is colder than

yesterday (q)

We will go swimming (r) only if

it is sunny

If we do not go swimming, then

we will take a canoe trip (s)

If we take a canoe trip, then we

will be home by sunset (t)

We will be home by sunset (t)

Definition

Fallacies (ngöy bi»n) resemble rules of inference but are

based on contingencies rather than tautologies

Example

If you do correctly every questions in mid-term exam, you will

get 10 grade You got 10 grade

Therefore, you did correctly every questions in mid-term

exam Is [(p ! q) ^ q] ! p a tautology?

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

Rules of Inference for Quantified Statements

Rule of Inference

8xP (x)) P (c)

P (c)for an arbitrary c) 8xP (x)

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

A student in this class has not gone to class

Everyone in this class passed the first exam

Someone who passed the first exam has not gone to class

Hint

C(x): x is in this class

B(x): x has gone to class

P (x): x passed the first

exam Premises???

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

1 9x(C(x) ^ :B(x)) Premise

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Given the predicate p(x) :00 x2 3x + 2 = 000 What is the

truth value (ch¥n trà) of the following propositions:

Contents Predicate Logic Exercise

Let x; y 2 Z+, and the predicate: p(x; y): "x is a divisor of y"

Determine the truth value of the following propositions:

Contents Predicate Logic

Exercise

O(x; y) : x is elder than y.

Express each of these statements using predicates:

grandfather, maternal grandmother’

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

9x9y(S(x; m) ^ O(x; m) ^ B(y; m) ^ :O(y; m))

8x(B(x; m) ! :O(x; m))

9x8y H(x; Thuyen) ^ H(y; Thuyen) ! (x = y)

or 9x8y H(x; Thuyen) ^ (x 6= y) ! :H(y; Thuyen)

9x8y(S(x; m) ^ :O(x; m) ^ S(y; m) ^ (x 6= y) ! O(y; m))

Contents

Predicate Logic

Exercise

Translating the following nested quantifiers:

Contents Predicate Logic Exercise

c is a brother (elder/younger) of m

If c is a brother of m and a is a father of m, then a is

elder than c and a is the father of c

Whoever is the sister of m, then c is also a brother of

that person

There is a sister of m or c is her husband, or there is a

husband of m and elder than m

All of the sisters of m are older or younger together

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Given a predicate N(x) "x has been to Da Lat" with the

domain is the all students in Mathematics class Translate the

following predicates into English

a)There is a student in this class has been to Da Lat.

b)All students in Math class have been to Da Lat.

c)There is no exists a student in Math class has gone to Da Lat.

d)There is a student in this class has never gone to Da Lat.

f) All students

in Math class have never been to

Da Lat.

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Given the predicate N(x) "x studies more than 5 hours in

class every weekday" with the domain is the all students in

Mathematics class Express the following predicates:

b)All of the students in Math class study over 5 hours every weekday.

c)There is a student who does not study in the class over 5 hours every

weekday.

d)There are no student studies in the class over 5 hours every weekday.

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic Exercise

H¢y cho bi‚t cæng thøc và tł cıa o⁄n m¢ gi£ (pseudo code) sau:

for (i = 0; i<numObjects; i++) {

There are no mushrooms that are poisonous and purple

8xM ushroom(x) ! :(P oisonous(x) ^ P urple(x))

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

H¢y cho bi‚t cæng thøc và tł cıa o⁄n m¢ gi£ (pseudo code) sau:

for (i=0; i<numObjects; i++) {

There is a mushroom that is purple and poisonous

9xM ushroom(x) ^ P oisonous(x) ^ P urple(x)

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents Predicate Logic Exercise

Cho o⁄n m¢ gi£ (pseudo code) sau:

// Look for first match

for (x=0; x<numKids; x++)

if isParent(Peter, kids[x])

match1Found = true;

// Now look for a second match

for (y=0; (y<numKids)&&(y!=x); y++)

if isParent(Peter, kids[y])

match2Found = true;

return match1Found && match2Found;

Bi‚t r‹ng: M£ng kids gçm 3 phƒn tß: { Alice, Bob, Charles } v

Peter ch¿ câ 1 con l Alice

H¢y cho bi‚t cæng thøc và tł cıa c¥u "Peter câ ‰t nh§t 2 con"

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Contents Predicate Logic Exercise

Cho P(x) l c¥u "x nâi ÷æc ti‚ng Nga" v Q(x) l c¥u "x bi‚t

a)Câ mºt sinh vi¶n ð tr÷íng b⁄n nâi ÷æc ti‚ng Nga v bi‚t Java.

b)Câ mºt sinh vi¶n ð tr÷íng b⁄n nâi ÷æc ti‚ng Nga nh÷ng khæng bi‚t Java.

c)Måi sinh vi¶n ð tr÷íng b⁄n •u nâi ÷æc ti‚ng Nga ho°c bi‚t Java.

d)Khæng câ mºt sinh vi¶n n o ð tr÷íng b⁄n nâi ÷æc ti‚ng Nga ho°c bi‚t

Contents

Predicate Logic

Exercise

Cho L(x,y) l c¥u "x y¶u y", vîi khæng gian cıa c£ x v y l t“p

hæp måi ng÷íi tr¶n th‚ giîi H¢y dòng c¡c l÷æng tł ” di„n ⁄t c¡c

c¥u sau

a)Måi ng÷íi •u y¶u Jerry.

b)Måi ng÷íi •u y¶u mºt ai â.

c)Câ mºt ng÷íi m t§t c£ måi ng÷íi •u y¶u.

d)Khæng câ ai y¶u t§t c£ måi ng÷íi.

e)Câ mºt ng÷íi m Lydia khæng y¶u.

f)Câ mºt ng÷íi m khæng ai y¶u.

g)Câ óng mºt ng÷íi m t§t c£ måi ng÷íi •u y¶u.

h)Câ óng hai ng÷íi m Lynn y¶u.

i)Måi ng÷íi •u y¶u ch‰nh m…nh.

j) Câ mºt ng÷íi n o â khæng y¶u ai ngo i ch‰nh m…nh.

z = y)))

x)j) 9x8y(L(x; y) ! x = y)

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

V… câ nhi•u c¡ch di„n ⁄t mºt cæng thøc và tł d÷îi d⁄ng ngæn

ngœ tü nhi¶n v sau ¥y l mºt c¡ch

÷ìng vîi x khæng d„

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Contents Predicate Logic Exercise

Dàch c¡c b£n mæ t£ sau ¥y sang ti‚ng Vi»t trong â F (p) l

b)N‚u t§t c£ m¡y in •u ang b“n in t i li»u th… s‡ câ t i li»u ang æi ÷æc in.

c)N‚u mºt t i li»u ang ÷æc æi in nh÷ng l⁄i bà m§t, chøng tä câ m¡y in n o â

¢ bà häng.

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

Chuy”n c¡c c¥u sau sang và tł, l÷æng tł v to¡n tß logic:

a)Khæng câ ai l ho n h£o.

b)Khæng ph£i måi ng÷íi •u ho n h£o.

c)T§t c£ b⁄n b– cıa b⁄n •u ho n h£o.

d)t nh§t câ mºt øa b⁄n cıa b⁄n l ho n h£o.

e)Måi ng÷íi •u l b⁄n cıa b⁄n v hå ho n h£o.

f)Khæng ph£i t§t c£ måi ng÷íi l b⁄n cıa b⁄n ho°c câ ai â khæng ho n h£o.

=) :8xD(x) _9yC(y)

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise

M»nh • và tł n o sau bi”u di„n c¥u : "Måi ch÷ìng tr…nh o t⁄o n‚u

câ möc ti¶u giŁng mºt ch÷ìng tr…nh kh¡c ¢ ¡p øng chu'n ABET

v k‚t qu£ ƒu ra câ th” ki”m chøng ÷æc th… công tu¥n theo

Contents Predicate Logic Exercise

Trong c¥u häi n y gi£ sß c¡c và tł:

Contents Predicate Logic Exercise

Chuy”n c¡c c¥u sau sang và tł, l÷æng tł v to¡n tß logic:

b¡o s‡ ÷æc gßi tîi måi ng÷íi dòng

ang «ng nh“p v o h» thŁng

bº nhî v tŁc º ÷íng tuy•n tŁi thi”u l 56 kbits/s

gh†t æng ta

Nguyen An Khuong, Tran Tuan Anh, Le Hong Trang

Contents

Predicate Logic

Exercise