chapter1 presentation (1)

8/10/2019 Chapter1 Presentation (1)

1/83

University of Reading 2010 www.reading.ac.uk

School of Systems Engineering

Discrete Mathematics

Chapter 1 SetsIntroduction


2/83

Sets

e!re going to start our "oo# at discretemathematics $ith the %asics of Set Theory&

Many courses "i#e this $i"" start at the

fundamenta" princip"es of "ogic' and %ui"d upto$ards set theory& e!re going to go the other$ay around&


3/83

Sets

e "oo# at sets (rst %ecause they!re a muchmore straightfor$ard p"ace to start in terms of$rapping your %rain around something&

Many $ou"d say that much of the "anguage ofset theory is )ust a particu"ar specia"isation ofthe "anguage of "ogic& *his is pro%a%"y accurate'%ut it is a"so the case that the "anguage of "ogic

is a genera"isation of the "anguage of set theory&So "et!s start $ith the easier one and go fromthere&


4/83

Sets

e!"" start %y setting out the %asic de(nitions ofsets and $ays to de(ne and descri%e them&

e "oo# at $hen t$o sets are e+ua" and $hat is

meant %y a ,su%set- of a set& e then "oo# at ho$ to ma#e ne$ sets from

ones $e #no$ a"ready&

e then (nd out a%out the four ma)or set

operations and c"ose %y "earning a%out using.enn Diagrams and mem%ership ta%"es&


5/83

Sets

*he #ey thing a%out set theory is that it givesus a "anguage and set of too"s $ith $hich $ecan treat co""ections of things as individua"

things& Indeed' set theory' is )ust the set of theoretica"

constructs and mathematica" de(nitions andprocesses re"ating to sets&


6/83




Chapter 1 Sets/art asic de(nitions


7/83

Sets

De(nition

set is an unordered co""ection of o%)ects&

*he o%)ects in a set are ca""ed the e"ements or

mem%ers of the set& set is said to contain its mem%ers&


8/83

Sets

otation

e $i"" usua""y use uppercase "etters torepresent sets&

*he sym%o" 3 means ,is an e"ement of-& *he sym%o" 4 means ,is 5* an e"ement of-&

e often "ist the e"ements of sets inparentheses&

e ca"" this the list method. hen a pattern is o%vious $e might use

ellipses6three dots together ,&&&-7 to indicatethat the pattern continues&


9/83

Sets

Example

e can de(ne the setsA' B, Cas8

A9 :1' 2' ;' ' ?@ B9 :1' ;' =' ?@

C9 :2'


10/83

Sets

e a"so use a more mathematica" set buildernotation to represent sets&

*hese usua""y use a varia%"e' oftenx' and put

into notation a sentence "i#e8 *his is the set of a""xfor $hichxis a positive integer$hich is "ess than 100&

Examples

:xBxis a positive integer andx 100@ or :x3 +Bx 100@


11/83

Eamous sets of num%ers

is the set of integers6$ho"e num%ers7 9 :&&&'F2'F1'0'1'2'&&&@

+is the set ofpositive integers +9 :1'2';'


12/83

Sets

Examples

e can re$rite the setsA' B, Cin set %ui"dernotation8

A9 :1' 2' ;' ' ?@9 :x3 B 1 x ?@

B9 :1' ;' =' ?@

9 :x3 Bxis odd and 1 x J@

C9 :2'


13/83

Revie$

hat is a setL

hat do $e ca"" the things in a setL

Can you give eamp"es of a set descri%ed %y

the "ist method and %y the set %ui"der methodL


14/83


15/83

Nercises

riteAand Bin set %ui"der and Cand Din "istmethod notations &

A 9:1';'='?'J@

9:x3 Bxis odd and 1 x J@

B 9:1''&&&@

9 :x3 Bx = k2for some k3 @

Is F>< 3 BL 5


16/83

Nercises

riteAand Cin set %ui"der and Band Din "istmethod notations &

C 9:x3 Bxis even@

9 :K' F' J' 12' 1=' 1A@

Is J 3 DL ONS Is 2? 3 CL 5


17/83 University of Reading 2010 www.reading.ac.uk



Chapter 1 Sets/art Su%sets and the emptyset


18/83


19/83


20/83

Repeated e"ements

ctua""y' $e $ou"d never specify a set %y$riting :1';'1'1' times in succession&

So $e ,got- :1';'1'1'


21/83

Sets in $hich thee"ements are sets

ote that there is nothing $rong $ith having setsthat have sets as their mem%ers&

Example

:S B S is one of the famous sets of num%ers from/art @

*his is the set :' +

' G' H@&


22/83

*he empty set

De(nition

*he empty set6or null set7is the set thatcontains no e"ements&

otation

e use the sym%o" to represent the emptyset&

Oou may see some %oo#s use :@ as $e""&

ote that and :@ are 5* the same


23/83

Su%sets

De(nition

*he setAis said to %e a subsetof a set B$henevery e"ement ofAis a"so an e"ement of B&

otation

e $riteAT B$henAis a su%set of B&

ExampleIf X9 :1' 2' ;' ' ?@ and Y9 :1' ;' =' ?@

then YTX.


24/83

/roper su%sets

De(nition

e sayAis aproper subset of B$henA T Band A B&

otation

e $riteA V B to denote thatAis a propersu%set of B.


25/83

Einite or in(niteL

De(nitions and otations

If a set has distinct e"ements $here is anonFnegative integer' $e say that the set is

fnite. *he num%er of e"ements in a (nite set' S' is

ca""ed the cardinalityof Sand is denoted %y BSB.

If a set is not (nite $e say it isinfnite.


26/83

Revie$

hen are t$o sets e+ua"L

hat is a su%setL

hat is a proper su%setL

hat is the empty setL hat is the cardina"ity of a setL


27/83

Nercise

fter $atching the rest of this video thin# a%outthese +uestions&

Eor any setAisATA L

Eor any setA !s TA L


28/83

Nercise


Eor any setAisATA L ONS& Nvery e"ement ofAis a"so inA67 and soAis a

su%set of itse"f&

Eor any setA !s TA L ONS *his one is $eird& ut it is true that ,every

e"ement of is a"so inA- even though there are ,no-

e"ements in &


29/83

Nercise


Eor any setAisAVA L

Eor any setA !s VA L


30/83

Nercise


Eor any setAisAVA L o& *his is the $ho"e point of a proper su%set&A is

5*a proper su%set of itse"f&

Eor any setA !s VA L ONS& s "ong as WW is not empty itse"f& *hen ,every

e"ement of is a"so inA- even though there are ,no-

e"ements in & nd is certain"y not e+ua" toA&


31/83

Nercises

fter $atching the rest of this video' thin# a%outtheseK

hat is BBL

IfX9 :x3 +

Bx is even andx 200@ $hat is BXBL

Come up $ith three eamp"es of an in(nite set&


32/83

Nercises

hat is BBL *he empty set has no mem%ers& So BB 9 0&


33/83

Nercises

IfX9 :x3 +Bx is even andx 200@ $hat is BXBL

*his is a"" the even num%ers "ess than 200& So it has to %e a%out 100K $e )ust have to %e carefu"

at the end of the set& X9 :2'


34/83

Nercises

Come up $ith three eamp"es of an in(nite set& Nasy ans$ers are ' G and H& ut you cou"d %ui"d )ust a%out anything and it $ou"d

count8

:1'


35/83




Chapter 1 Sets/art C Sets from sets


36/83

Sets from sets

e!re no$ going to "oo# at t$o $ays to create ane$ set from one' or more' origina" sets&

ExampleI ma#e my hote" %rea#fast %y choosing8 5ne )uice from :orange' grapefruit' tomato@ 5ne cerea" from :cornYa#es' %ranYa#es' cocopops@ *hree hot items from

:eggs' sausage' %eans' hash %ro$ns' mushrooms'%acon@

*oast from :$hite' %ro$n@ Drin# from :tea' coZee@


37/83


38/83

Namp"e

Eind the po$er set of S9 :1'2';@&


39/83

Namp"e


:' :1@' :2@' :;@' :1'2@' :1';@' :2';@':1'2';@ @


40/83


41/83

Cartesian product

De(nition

*he Cartesian product of a pair of setsAand Bis the set of a"" ordered pairs 6a' $7 $here a 3A

and$

3B

&

otation

e use the notationA [ B to denote the

Cartesian product ofAand B&

A [ B = : 6a, $7 B a 3A and $3 B@


42/83

nFtup"es

De(nition

n ordered n-tupleis a "ist of e"ements inorder that $e may $rite as 6a1, a2, a;, ..., an#.

n ordered pair cou"d %e ca""ed an ordered 2Ftup"e&


43/83

Cartesian product

De(nition

*he Cartesian product of a co""ection of setsA1'A2'&&&'A is the set of a"" ordered Ftup"es

6a1, a2, a;, ..., an#$here a%3A1' a23A2' &&& ' a3An&

A1[ A2 [ &&& [A9: 6a1, a2, a;, ..., an#B a!3Ai'

for !9 1'2,...,@


44/83


45/83

Sets from sets

ExampleI ma#e my hote" %rea#fast %y choosing8

5ne )uice fromA9 :orange' grapefruit' tomato@ 5ne cerea" from B9 :cornYa#es' %ranYa#es'

cocopops@ *hree hot items from

C9 :eggs' sausage' %eans' hash %ro$ns' mushrooms'%acon@

*oast from D9 :$hite' %ro$n@

Drin# from E9 :tea' coZee@

*he set of possi%"e %rea#fasts is8

A [ B [ C [ C[ C[ D[ E 9 A [ B [ C;[ D

[ E


46/83

Revie$

hat is the po$er set of a setL

hat is the Cartesian product of a pair of setsL

hat is the Cartesian product of a co""ection ofsetsL

You should now be ready to try thesuggested Exercises at the end of Section2. in the book.


47/83

Namp"e and Nercises

e!"" "oo# at this +uestion no$K


*ry these after $atching the video& hat is the po$er set of the empty setL

hat is the po$er set of :a'%'c@L Po$ is this re"ated to your ans$er to the (rst

+uestionL hat can you say from this a%out the nature and

cardina"ity of the po$er set of any set $ith ;e"ementsL

Can you etrapo"ate this and $or# out the cardina"ityof the po$er set of any set $ith e"ementsL


48/83


49/83

Nercises


:' :1@' :2@' :;@' :1'2@' :1';@' :2';@':1'2';@ @

hat is the po$er set of :a'%'c@L

:' :a@' :%@' :c@' :a'%@' :a'c@' :%'c@':a'%'c@ @

*hese sets are %asica""y the same as one another' )ust$ith 1 ,mapped to- a' 2 mapped to %' ; mapped to c& In fact' the po$er set of any set $ith ; e"ements $i""

"oo# eact"y "i#e this' and have A very simi"are"ements&

*he cardina"ity of the po$er set of any set $ith e"ements is )ust 2L


50/83

Nercises

*ry these after $atching the video&

\etA9 :1'2';@ and B9 :='>@ and C9 :p'&@&

Eind the fo""o$ing sets& A [A A [ B [ C

IfX9 :1'2@ and Y9 ' $hat isX [ Y L

If P [ Q = $hat must %e trueL


51/83


52/83


53/83

S h l f S t E i i


54/83




Chapter 1 Sets/art D Set operations


55/83

Set operations

e have to "earn four ma)or set operations&

e $i"" do this %y "oo#ing at .enn Diagrams andmem%ership ta%"es&


56/83


57/83

.enn diagrams


58/83


59/83


60/83


61/83


62/83


63/83

Union

otation *he unionof t$o sets'Aand B' is denoted %yA

_ B&


64/83


65/83

Comp"ement

otation *he complementof a setAis denoted %y' 6or

sometimes %yA( 7&

ote that you don!t need to have t$o sets tode(ne the comp"ement&


66/83

DiZerence

otation *he dierenceof t$o sets'Aand B' is denoted

%yA B&


67/83


68/83


69/83

Nercises

De(ne each of the four operations using set%ui"der notation&

hat $ou"d a .enn diagram "oo# "i#e sho$ingt$o sets for $hichA V B ) that !s, A !s a propers*$set of B L

rite do$n epressions in terms of operationson sets for the co"oured areas %e"o$&



70/83



Chapter 1 Sets/art N Mem%ership ta%"es


71/83


72/83

Mem%ership ta%"es

Example

+e o ot ormally la$el the left ha s!e -!thp, & , r, s.

Th!s !s *st to sho- yo* ho- !t -orks.

A B

p 1 1

+ 1 0

r 0 1

s 0 0


73/83

Namp"e

Comp"ete the mem%ership ta%"e sho$n here&

A B A ! B

1 1

1 0

0 1

0 0


74/83


75/83


76/83


77/83

Nercise

Comp"ete the mem%ership ta%"e %e"o$&

hat do you noticeL

hat does this meanL

A B A " B A # (A "B)

1 1

1 0

0 1

0 0


78/83

Nercise

Comp"ete the mem%ership ta%"e %e"o$&

*he right co"umn is a"" `eros&

*he set is empty&

A B A " B A # (A "B)

1 1 1 0

1 0 1 0

0 1 1 0

0 0 0 0


79/83

Nercise

Comp"ete the fo""o$ing mem%ership ta%"e&

A B A " B $A " B%"

B " A " $B "%

1 1 1

1 1 0

1 0 1

1 0 0

0 1 1

0 1 0

0 0 1

0 0 0


80/83

Nercise

Comp"ete the fo""o$ing mem%ership ta%"e&

A B A " B $A " B%"

B " A " $B "%

1 1 1 1 1 1 1

1 1 0 1 1 1 1

1 0 1 1 1 1 1

1 0 0 1 1 0 1

0 1 1 1 1 1 10 1 0 1 1 1 1

0 0 1 0 1 1 1

0 0 0 0 0 0 0


81/83


82/83

Nercise

Use mem%ership ta%"es to verify the fo""o$ing&

A B A " B A ! (A "

B)1 1 1 1

1 0 1 1

0 1 1 0

0 0 0 0


83/83

Nercise

Use mem%ership ta%"es to verify the fo""o$ing&

A B B " A ! $B"

%

A ! B A !

(A ! B% " $A !

%

1 1 1 1 1 1 1 1

1 1 0 1 1 1 0 1

1 0 1 1 1 0 1 1

1 0 0 0 0 0 0 00 1 1 1 0 0 0 0

0 1 0 1 0 0 0 0

0 0 1 1 0 0 0 0

chapter1 presentation (1)

Documents