sets real numbers and operations on real numbers

8/19/2019 Sets Real Numbers and Operations on Real Numbers

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Basic concepts


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Example 1◦ Set of Books in the house

◦ Set of iPhone Apps

◦ Set of numbers from 1 to 100 which

contains the number one “1”

Denition 1 Set ! A collection of ob"ects or numbers


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Denition # Elements ! $b"ects or %umbers in a set


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& ' ! (on)ention %otation use* to *enote the collectionof elements+ ,he elements are usuall- separate* b-commas+

(apital .etters! ,-picall- use* for the name of a set


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Example /◦ A = {1, 2, 3}◦

The set of Apps in the iPhone/iTouch◦ Population of all humans inhabiting the earth

Example ◦ Set of all counting numbers◦ Set of all hole numbers◦ Set of all rational numbers

Denition / inite set ! A set that has a xe* number ofelements+

Denition 2nnite set ! A set without a xe* number ofelements+


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◦ Example ! = {1, 2, 3, """} # this set continues to in$nit%

A = {1, 2, 3,""", &'} # this set is $nite an( stops at &' B = {1, 3, &, ), """,} # set of o(( counting numbers

◦ Example

*or the gi+en set B = {1, 2, 3, """ , -} list this in +ariable form B = {. . is a natural number less than &'}

+++ ! 2n*icates a continuin3 pattern in a set

4ariable ! 5se* to stan*6represent for some numbers+5suall- *enote* b- letters+

7e can use )ariables to represent the numbers in a set8 9 ,his notation is actuall- the phrase “such that”


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◦ 0.ample 1 B is rea( as

1 is a member of B, 1 is an element of B 1 is in B

◦ 0.ample 1 B is rea( as

1 is not a member of B 1 is not an element of B 1 is not in B

! use* to in*icate that a specic number6ob"ect is a member ofa set

! use* to in*icate that a specic number6ob"ect is a member ofa set


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◦ 0.ample &

0ual Sets {3, , )} = {3, , )} an( {2, , 1} = {1, 2, }

!on4eual sets {3, &, 5} = {3, &, )}

Denition : E;ualit- of Sets 9 To sets are eual if the% contain e.actl%the same members" 6therise, the% are sai( to be not eual

= ! 2n*icates e;ual sets

= ! 2n*icates that sets are not e;ual


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◦ 0.ample 5 A = {1, 2, 3, }

B = {, &, 5, )}

A B = {1, 2, 3, , &, 5, )}

Denition


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◦ 0.ample ) A = {1, 2, 3, }

B = {, &, 5, )}

A B = {}

Denition = 2ntersection of Sets7f A an( B are sets, the intersection of A an( B, (enote( AB, is the set of all elements that are in both A an( B" 7ns%mbols,

A B ={.. A an( . B}


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%ote that A > A an* A > for an- set A+

Denition ? Empt- SetA set ith no members (enote( b% the s%mbol

%ote that the set &0' is not the empt- set+


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2f A is not a subset of B@ we write A B+

Example◦

@/' @/@'@ &1@#@/@@:'&1@#@/@@:@


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The Set of 8eal !umbers an(7ts Subsets


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!atural 9:ounting; !umbers! = {1,2,3,"""}


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Another a% to (escribe 8ational !umbersis b% using their (ecimal form"

ational numbers are those *ecimal numbers whose *i3itseither repeat or terminate


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?raphing on the !umber @ine◦ 1st step (ra a straight line an( label an%

con+enient point ith the number '"

◦ 2n( step choose an% con+enient length an( useit to locate points to the right of ' 9positi+eintegers; an( to the left of ' 9negati+e integers;

'

' 1 2 3 & 5 )41424344&454)4


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?raphing on the !umber @ine

0 1 # / : < =919#9/99:9


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?raphing on the !umber @ine 7t is often con+enient to illustrate sets of numbers

on a number line" The set of 7ntegers is illustrate(belo

0.ample

◦ Tr% to plot the set of counting numbers on a

graphing line

0 1 # / 919#9/9 ++++++• • • • • • • • •

0 1 # / 919#9/9 ++++++• • • •


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◦ 1"1213&52""" # neither terminating nor repatingthus is an irrational number"

◦ ◦ '"5'5'''5'''''5'''''''5"""

◦ '"1&11&111&1111&"""

◦ 3"123&5)-1'111213"""

◦

,he 2rrational %umbers # numbers hich cannot be e.presse(as a ratio of integers" !either terminating nor repeating"


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The set of real numbers can be +isualiDe( asthe set of all points on the number line"

0 1

9

# /9/ 9# 91• • • • • • •

16#9#+ 916/

++++++

The set of 8ational an( 7rrational numbers ha+e no numbers incommon an( together form the set of real numbers "


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•

8eal !umbers

7rrational !umbers8ational !umbers

7ntegers


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0.amples◦ 92,3; # set of real numbers that lie beteen 2 an( 3 on the

number line 9(oesnEt inclu(e 2 an( 3;"

◦ F2,3G # set of real numbers that lie beteen 2 an( 3 on thenumber line 9inclu(es 2 an( 3;"

◦

C9 @ # set of all real numbers

2nter)al %otation # 9a,b; or Fa,bG here a an( b are the to

en(points of the inter+al"

Ce$nition 1' 2nter)als of eal %umbers # set of real numbers that liebeteen to real numbers hich are calle( the en(points of the inter+al"

0 1 # / 919#9/9 ++++++

o o

0 1 # / 919#9/9 ++++++• •


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6perations on the set of 8eal!umbers


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7n algebra, computations are no performe( ith positi+e an( negati+enumbers" Basic operations of arithmetic are exten*e* to ne3ati)enumbers"


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◦ 0.amples 8:8 > :

89:8 > :+

Absolute 4alue of a number # the numberEs (istance from ' onthe number line"

S%mbol for absolute +alue of a # 8a8

Absolute +alue represents (istance an( this is ne+er negati+e" Thus 8a8 > '"

' 1 2 3 & 5 )41424344&454)4

: units

$ri3in

: units


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0.ample◦

& an( 4& are opposites of each other"

◦


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0.ample◦ 4) = 494); = )"

Ce$nition 12 Absolute 4alue #*or an% number a,


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A((ition

45 H 94); = I

= 45 H 4) ?et the absolute +alue of both numbers = 4945; H 494); Ce$nition of absolute +alue

= 5 H ) Ce$nition of an opposite

= 13 Basic a((ition

= 413 B% Ce$nition 13

Ce$nition 13 Sum of ,wo %umbers with .ike Si3ns To $n( the sum of to numbers ith the same sign, a(( their absolute+alues" The sum has the same sign as the original numbers"

The number a an( its opposites #a ha+e a sum of Dero for an%letter a" a an( #a are calle( a**iti)e in)erses of each other"

Ce$nition 1 A**iti)e 2n)erse Propert-*or an% real number a, there is a uniue number #a such that

a H 94a; = 4a H a = 0"


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0.ample◦ 4) H 1' = I

◦ = 4) 4 1' ?et the absolute +alue of both numbers

◦ = ) # 1' Subtract them from each other

◦ =43 B% Ce$nition 1& the number ith the largerabsolute +alue is 1'

◦ > /

Ce$nition 1& Sum of ,wo %umbers with 5nlike Si3ns Can**iFerent absolute )alues

To $n( the sum of to numbers ith unliJe signs, subtract theirabsolute +alues"

The sum is positi+e if the number ith the larger absolute +alue is positi+e The sum is negati+e if the number ith the larger absolute +alue is negati+e"


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Subtraction

4 ) # 3 = I = 4) H 943; B% Ce$nition 15 = ) H 3 B% Ce$nition 13 = 1' Basic a((ition

>910 B- Denition 1/

Ce$nition 15 Subtraction of eal %umbers*or an% real numbers a an( b,

a # b = a H 94b;


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Kultiplication

Pro*uct of to numbers # result of multiplication of to numbers" Thenumbers multiplie( are calle( factors"

The pro(uct of a an( b is ritten as a Gb or ab"

10@ so we ha)e :C# > 10


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Di)ision

0.ample◦ The reciprocal of 3 is 916/"◦ 3 N91/3; = 1

@iJe a((iti+e in+erses, e+er% nonDero real number a has a multiplicati+ein+erse or reciprocal 91/a;"

Ce$nition 1 Iultiplicati)e 2n)erse Propert-*or an% nonDero real number a, there is a uniue number 1/asuch that

7f the number is negati+e then its reciprocal is also negati+e"


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Di)ision

0.ample◦ =

◦ =

◦ = 2

Ce$nition 1- Di)ision of eal %umbers*or an% real numbers a an( b ith b = ',

a # (i+i(en( b # (i+isor c # uotientO or is also calle( the uotient


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Di)ision b- Jero 7f e rite , e nee( to $n( c suchthat

But there is no such number an( it ill be confusing" Thus

is (e$ne( onl% for b = '"

are sai( to be un(e$ne("


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0+aluating 0.pressions


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7n algebra %ou ill learn to orJ ith +ariables" oe+er, there is oftennothing more important than $n(ing a numerical anser to a uestion


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◦ Example >: H C#C/ # in+ol+es more than one operation of arithmetic

>: H C11

>8 9=H 89= H 9#8

>89/18 9 ,he absoulte )alue is a 3roupin3 s-mbol aswell

>/1

Arithmetic Expression # The result of riting numbers in a meaningfulcombination ith the or(inar% operations of arithmetic

C ! Parentheses are use* as 3roupin3 s-mbols to in*icate whichoperations are performe* rst+ K L ! Brackets are also use* to

in*icate 3roupin3


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Exponents ! %otation use* to simplif- the writin3 of a repeate*multiplication

Ce$nition 2' Exponential Expression

*or an% natural number n an( real number a,


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7e use the ra*ical s-mbol to in*icate the nonne3ati)e orprincipal s;uare root of a number+

Ce$nition 21 S;uare oots7f , then a is calle( a s;uare root of b" 7f , then ais calle( the principal s;uare root of b an( e rite+

,he ra*ical s-mbol is a 3roupin3 s-mbol+ 7e perform alloperations within the ra*ical s-mbol before the s;uare root isfoun*+


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Example

0+aluate 91' # ; an( 954; $rst

Then (i+i(e 2 b% 42

91

7hen an expression in)ol)es a fraction bar@ the numerator an**enominator are each treate* as if the- are in parentheses+


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Ce$nition 22 $r*er of $perations0+aluate insi(e an% grouping s%mbols $rst+


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Properties of 8eal !umbers


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Example : H / > / H :

? > ?

Ce$nition 23 (ommutati)e Propert- of A**ition*or an% real numbers a an( b,

aHb > bH a+

(ommutati)e Propert- of Iultiplication*or an% real numbers a an( b,

ab = ba"

Subtraction an* Di)ision are not commutati)e operations+

:C /0


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Example

C H : H ? > H C: H ? C H ? > H C1/

1= > 1=

Ce$nition 2 Associati)e Propert- of A**ition*or an% real numbers a, b an( c,

CaHb Hc > a H Cb H c+

Associati)e Propert- of Iultiplication*or an% real numbers a, b an( c,

9ab;c = a9bc;"

#C/C: >C#C/C:#C1: > /0


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/Cx9# > /Cx H C9# Denition of Subtraction of eal%umbers

> /x H C9 /x 9 ab H ac+


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Ce$nition 25A**iti)e 2*entit- Propert-*or an% real number a,

aH0 > 0 H a > a+

Iultiplicati)e 2*entit- Propert-*or an% real number a,

a C1 > 1 Ca > a+


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Ce$nition 2)A**iti)e 2n)erse Propert-

*or an% real number a, there is a uniue number # a suchthat

aHC9a > 9a H a > 0+

Iultiplicati)e 2*entit- Propert-*or an% real number a, there is a uniue number 91/a; such that

a C16a > 16a Ca > 1+


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Ce$nition 2 Iultiplication Propert- of Jero*or an% real number a,

0 Ca > a C0 > 0+


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Exercises◦ .ist the elements in each set 1 A> &x8x is an e)en natural number less than

#0'

# B> &x8x is an o** natural number less than1'

◦ .ist usin3 )ariable notation 1 &1@ #@ /@ @ :@


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Exercises◦ 5sin3 the sets A@ B@ (@ an* %+ Determine

whether each statement is true or false+

◦ A > &1@ /@ :@ =@ '

◦ B > @ @ &1@ #@ /@ +++' 1 5 B

# B

/ :=!

!=A


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Exercises◦ 5sin3 the sets A@ B@ (@ an* %@ list the

elements in each set+ 2f the set is empt-write + See Examples # an* /+

◦

A > &1@ /@ :@ =@ '◦ B > @ @ &1@ #@ /@ +++' 1 A ( > O

# A B > O

/ A B > O

A > O


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Exercises◦ Determine whether each statement is true

or false+ Explain -our answer+

◦ A > &1@ /@ :@ =@ '

◦ B > @ @ &1@ #@ /@ +++' 1 A % > O

# B ( > O

/ B > O

( A > O


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Exercises◦ Determine whether each statement is true

or false+ Explain -our answer+

1 2s 9< an element of the set of ational%umbersO

# 2s the set of %atural numbers a subset of theset of ational %umbersO


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Exercises◦ .ist the elements in each set an* 3raph

each set on a number line+

1 &x8 x is a whole number smaller than


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Exercises◦ 7rite each inter)al of real numbers in

inter)al notation an* 3raph it+

1 ,he set of real numbers 3reater than 1 # ,he set of real numbers between 0 an* #

inclusi)e

/ ,he set of real numbers 3reater than ore;ual to 1 an* less than /+


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Exercises◦ E)aluate+

1 89/8

# 808 / 89


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1 = ! 10 >

# 91 9 : > / 9 ! < >

#0 ! C9/ >


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1 #:C9/ >

# C:C9= > / :C: >

9< C9 >


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Exercises◦ in* the multiplicati)e in)erse of each

number

1 #0 > # 9: >

/ 9


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Exercises◦ E)aluate+ 2f a ;uotient is un*ene*@ sa- so+

1 >

# > / >


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sets real numbers and operations on real numbers

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