ds lecture 6

8/9/2019 Ds Lecture 6

1/30

(CSC 102)

Lecture 6

Discrete Structures


2/30

Previous Lectures Summary

•Different forms of arguments

•Modus Ponens and Modus Toens

• !dditiona "aid !rguments

•"aid !rgument #it$ %ase Concusion

•&nvaid argument #it$ a true Concusion

•Converse and &nverse error

•Contradictions and vaid arguments


3/30

Predicates and 'uantified statements &


4/30

Todays LectureTodays Lecture

• Predicates

• Set otation

• *niversa and +,istentia Statement

• Transating -et#een forma and informa anguage

• *niversa conditiona Statements

• +.uivaent %orm of *niversa and +,istentia

statements

• &m/icit 'uaification

• egations of *niversa and +,istentia statements


5/30

PredicatesPredicates

! /redicate is a sentence #$ic$ contains finite num-er ofvaria-es and -ecomes a statement #$en s/ecific vaues

are su-stituted for t$e varia-es

T$e domain of a /redicate varia-e is t$e set of a vaues t$atmay -e su-stituted in /ace of t$e varia-e

Truth Set

&f P(,) is a /redicate and , $as domain D t$e trut$ set of

P(,) is t$e set of a eements of D t$at mae P(,) true#$en su-stituted for , T$e trut$ set of P(,) is denoted -y

read as 3t$e set of a , in D suc$ t$at P(,)4

{ | ( )} x D P x∈


6/30

%or any t#o /redicates P(,) and '(,) t$e notation

means t$at every eement in t$e trut$ set of

P(,) is in t$e trut$ set of '(,) T$e notation

means t$at P and ' $ave identica trut$ sets

Consider t$e /redicate5

T$e trut$ set of t$e a-ove /redicate is

( ) ( ) P x Q x⇒

otation

( ) ( ) P x Q x⇔

R x x ∈> ,0

{ }0>∈ + x R x


7/30

Example

Let P(,) , is a factor of 7 '(,) , is a factor of 8

and 9(,) x : ; and T$e domain of , is

assumed to -e *se sym-os to indicate

true reations$i/s among P(,) '(,) and 9(,)

a T$e trut$ set of P(,) is


8/30

Cont?

Let Q(x, y) be the statement x + y = x − y

#$ere t$e domain for , and y is t$e set of a rea num-ers

Determine t$e trut$ vaue of5

(a) '(;@2)(-) '(8A 0)

(c) Determine t$e set of a /airs of num-ers , and y suc$ t$at

'(, y) is true

Solution:

(a) '(;@2) says t$at ; B (@2) ; @ (@2) or A #$ic$ is fase

(-) '(8A 0) says t$at 8AB 0 8A @ 0 #$ic$ is true

(c) , B y , @ y if and ony if , B 2y , #$ic$ is true if and ony if

y 0 T$erefore , can -e any rea num-er and y must -e ero


9/30

*niversa and +,istentia Statements

Let '(,) -e a /redicate and D t$e domain of , !

universa statement is of t$e form 3 4 &t is

true if and ony if '(,) is true for a , in D and it is

fase if and ony if '(,) is fase for at east one , in D !

vaue for , for #$ic$ '(,) is fase is caed acountere,am/e to t$e universa statement

+,am/e5 Let D


10/30

Cont?Cont?

>ence is true

T$e tec$ni.ue used in first statement #$ie s$o#ing t$e

trut$ness of t$e universa statement is caed met$od of

e,$austion

Consider t$e statement %ind t$e counter

e,am/e to s$o# t$at t$is statement is not true

Counter e,am/e Tae ,1E2 t$en , is in 9 and

>ence is fase

2, . x D x x∀ ∈ ≥

2, . x R x x∀ ∈ ≥

21 1

2 2

≤ ÷ ÷

2

, . x R x x∀ ∈ ≥


11/30

+,istentia 'uantifier

Let Q(x) -e a /redicate and D t$e domain of

, !n e,istentia statement is of t$e form

suc$ t$at

&t is true if and ony if '(,) is true for at east

one , in D &t is fase if and ony if '(,) is

fase for a , in D

T$e sym-o denotes 3t$ere e,ist4 and iscaed t$e e,istentia .uantifier ∃

D x ∈∃ )( xQ


12/30

Trut$ and fasity of +,istentia statements

Su//ose P(x) is the predicate “x < |x|.” Determine thetruth value ! x s.t. P(x) "here the dmain !r x is#∃

(a) t$e t$ree num-ers 1$ %$ &.

(-) t$e si, num-ers −%$−'$ $ '$ %$ &.

Solution

(a) P(')$ P(%)$ and P(&) are all !alse ecause in each

case x = |x|. *here!re$ x such that P(x) is !alse∃ !r

this domain

(-) &f #e -egin c$ecing t$e si, vaues of x$ "e !ind

P(−%) is true. t states that −% < |−%|. ,e need to c$ec

no furt$erF $aving one case t$at maes t$e /redicate

true is enoug$ to guarantee t$at x s.t. P(x) is true.∃


13/30

Trut$ and fasity of +,istentia statements

Consider t$e statement S$o#t$at t$is statement is true

So5 o-serve t$at T$us is true for at

east one integer m >ence is

true

Let +


14/30

Transating from forma to informa anguage

9e#rite t$e foo#ing statements in a variety ofe.uivaent -ut more informa #ays Do not use t$e

sym-o

a)

-)

c)

Soution5 a) #e can #rite t$e statement in many #aysie 3 ! rea num-ers $ave non negative s.uares4

3o rea num-er $as a negative s.uare4

3 , $as a non negative s.uare for eac$ vaue of ,4

,∀ ∃

2, 0. x R x

∀ ∈ ≥2, 1. x R x∀ ∈ ≠ −

mmt s Z m =∈∃ 2..


15/30

Cont….

-) Simiary #e can transate t$e second statement int$ese #ays

3 ! rea num-ers $ave s.uares not e.ua to H14

3o rea num-er $ave s.uare e.ua to H14

c) 3T$ere is an integer #$ose s.uare is e.ua to itsef 4

3#e can find at east one integer e.ua to its o#ns.uare4


16/30

Cont?

Write the following statement in English, using the

predicates

-(x)# “x is a -reshman”

* (x$ y)# “x is tain/ y”

#$ere x represents students and y represents curses#

∃ x ( -(x) * (x$ Discrete 0ath)∧ )

Solution

T$e statement x (-(x) * (x$ Discrete)) says that there∃ ∧

is a student x "ith t" prperties# x is a !reshman and x

is tain/ Discrete. n 1n/lish$ “2me -reshman is tain/

Discrete 0ath.”


17/30

Transating from informa Language to %orma anguage

3+very fres$man at t$e Coege is taing CSC 1024Solution: T$ere are various #ays to ans#er t$is .uestionde/ending on t$e domain

• &f #e tae as our domain a fres$men at t$e Coege

and use t$e /redicate * (x) # “x is tain/ 323 1024t$en t$e statement can -e #ritten as x$ *(x)∀ .

• Ie are maing a conditiona statement5

3&f t$e student is a fres$man t$en t$e student is taing

CSC 101F4

∀ x$ (-(x) 4 * (x)).

ote t$at #e cannot say x (-(x) * (x))$ ecause this∀ ∧

says that every student is a !reshman$ "hich is nt somet$in #e can assume $ere


18/30

Cont?Cont?

3+very fres$man at t$e Coege is taing some Com/uterScience course4

So5 &f #e tae as our domain for /eo/e a fres$men at

t$e Coege and our domain for courses a Com/uterScience courses

T$en #e can use t$e /redicate

* (x$ y)# “x is tain/ y”

T$e statement can -e #ritten as

∀ x y *(x$ y).∃


19/30

*niversa Conditiona Statements*niversa Conditiona Statements

! reasona-e argument can -e made t$at t$e most

im/ortant form of statement in mat$ematics is t$e

universa conditiona statement5

∀ x$ i! P(x) then Q(x)

1xample# 3+veryone #$o visited %rance stayed in

Paris4

So5 >o#ever if #e tae a /eo/e as t$e universe

t$en #e need to introduce t$e /redicate -(x) !r “x

visited %rance4 and P(x) is the predicate “x stayed in

Paris.” &n t$is case t$e /ro/osition can -e #ritten as

∀ x$ ( -(x) 4 P(x) ).


20/30

• Ie can #rite t$e foo#ing statements in avariety of informa #ays

if t$en

So5• if a rea num-er is greater t$en 2 t$en t$e

s.uare is greater t$an 8

• I$enever a rea num-er is greater t$en 2

its s.uare is greater t$an 8

• T$e s.uares of rea num-er greater t$an 2

are greater t$an 8

, x R∀ ∈ 2 x > 2 4 x >


21/30

+,ercise+,ercise

9e#rite t$e foo#ing statements in t$e form ∀ $i! then .

a) &f a rea num-er is an integer t$en it is a rationa

num-era) ! -ytes $ave eig$t -its

-) o fire trucs are green

So5 a)

-) x$ i! x is a yte$ then x has ei/ht its.∀

c). x$ i! x is a !ire truc$ then x is nt /reen.∀

, , . x R ifx Z thenx Q∀ ∈ ∈ ∈


22/30

+.uivaent %orms of *niversa and +,istentia statements

J-serve t$at t$e t#o statements 3 real∀numers x$ i! x is an inte/er then x is

ratinal ” and “ inte/ers x$ x is ratinal”∀

mean the same thin/.

&n fact a statement of t$e form

if P(,) t$en '(,)

Can a#ays -e re#ritten in t$e form

Can -e re#ritten as

∀, if , is in D t$en '(,)

, ( ) x D Q x∀ ∈

,U x ∈∀


23/30

Contd

T$e foo#ing statements are e.uivaent

∀ /oygons P if P is s.uare t$en P is a rectange !nd

∀ s.uares P P is a rectange

T$e e,istentia statements

∃ , -eongs to * suc$ t$at P(,) and '(,)

!nd

∃ , -eongs to D suc$ t$at '(,)

!re aso e.uivaent /rovided D is taen to consist of a eements in

* t$at mae P(,) true


24/30

+.uivaence form for e,istentia statement

T$e foo#ing statements are e.uivaent5

∃ a numer n such that n is prime and n is even

5nd

∃ a prime numer n such that n is even.


25/30

&m/icit 'uantifications&m/icit 'uantifications

• Consider 3 &f a num-er is an integer t$en it is arationa num-er 4

T$e cue to indicate its universa .uantifications comesfrom t$e /resence of t$e indefinite artice 3a4

+,istentia .uantification can aso -e im/icit

for instance 3 t$e num-er 28 can -e #ritten as a sum of

sum of t#o integers4“∃ even inte/ers m and n such that %6=m + n.”


26/30

T$e negation of t$e statement of t$e form ∀ , in D '(,)

is ogicay e.uivaent to a statement of t$e

form

∃ x in D such that 7Q(x)

2ymlically#

8te# the ne/atin ! universal statement is

l/ically e9uivalent t existential statement.

egations of 'uantified Statement

~ ( , ( )) ~ ( ). x D Q x x D Q x∀ ∈ ≡ ∃ ∈ ∋


27/30

T$e negation of t$e statement of t$e form ∃ x in D such that '(,)

is ogicay e.uivaent to a statement of t$e

form

∀ , in D 7Q(x)

2ymlically#

8te# the ne/atin ! existential statement is

l/ically e9uivalent t universal statement.

Cont?

~ ( ( )) , ~ ( ). x D Q x x D Q x∃ ∈ ∋ ≡ ∀ ∈


28/30

+,am/es

Negate Some integer x is positive and all integersy are negative.”

Solution: *sing a integers as t$e universe for x and y$

the statement is x s.t. (x : ) y$ (y < ).∃ ∧ ∀ *he

ne/atin is7; x (x : ) y (y < ) 7 x s.t. (x : ) 7 y$ (y s la"

∀ x$ 7(x : ) y s.t. 7(y < )∨ ∃ prperties ! ne/atin

∀ x$ (x ? ) y s.t. (y @ ).∨ ∃

T$erefore t$e negation is 3+very integer x is nn

psitive r there is an inte/er y that is nnne/ative.”


29/30

Cont?

Negate There is a student who came late to classand there is a student who is a!sent from class."

Solution: &n sym-os if A(x) # “x came late t class” and

5(x) # “x is asent !rm class$” this statement can -e

#ritten as x st A(x) y st 5(y).∃ ∧ ∃ote t$at #e must use a second varia-e y. By ne !

De 0r/an>s la"s the ne/atin can e "ritten as

7( x st A(x)) 7( y st 5(x)) x$ 7A(x) y$ 75(x).∃ ∨ ∃ ∀ ∨ ∀

&n +ngis$ t$is is 3o student came ate to cass or no

student is a-sent from cass4


30/30

Lecture SummaryLecture Summary

• Predicates

• Set otation

• *niversa and +,istentia Statement

• Transating -et#een forma and informa anguage

• *niversa conditiona Statements

• +.uivaent %orm

• &m/icit 'uaification

• egations

ds lecture 6

Documents