sets real numbers and operations on real numbers
TRANSCRIPT
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
1/64
Basic concepts
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
2/64
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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
3/64
Denition # Elements ! $b"ects or %umbers in a set
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
4/64
& ' ! (on)ention %otation use* to *enote the collectionof elements+ ,he elements are usuall- separate* b-commas+
(apital .etters! ,-picall- use* for the name of a set
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
5/64
Example /◦ A = {1, 2, 3}◦
The set of Apps in the iPhone/iTouch◦ Population of all humans inhabiting the earth
Example ◦ Set of all counting numbers◦ Set of all hole numbers◦ Set of all rational numbers
Denition / inite set ! A set that has a xe* number ofelements+
Denition 2nnite set ! A set without a xe* number ofelements+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
6/64
◦ Example ! = {1, 2, 3, """} # this set continues to in$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”
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
7/64
◦ 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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
8/64
◦ 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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
9/64
◦ 0.ample 5 A = {1, 2, 3, }
B = {, &, 5, )}
A B = {1, 2, 3, , &, 5, )}
Denition
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
10/64
◦ 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}
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
11/64
%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+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
12/64
2f A is not a subset of B@ we write A B+
Example◦
@/' @/@'@ &1@#@/@@:'&1@#@/@@:@
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
13/64
The Set of 8eal !umbers an(7ts Subsets
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
14/64
!atural 9:ounting; !umbers! = {1,2,3,"""}
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
15/64
Another a% to (escribe 8ational !umbersis b% using their (ecimal form"
ational numbers are those *ecimal numbers whose *i3itseither repeat or terminate
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
16/64
?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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
17/64
?raphing on the !umber @ine
0 1 # / : < =919#9/99:9
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
18/64
?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 ++++++• • • •
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
19/64
◦ 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"
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
20/64
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 "
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
21/64
•
8eal !umbers
7rrational !umbers8ational !umbers
7ntegers
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
22/64
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 ++++++• •
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
23/64
6perations on the set of 8eal!umbers
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
24/64
7n algebra, computations are no performe( ith positi+e an( negati+enumbers" Basic operations of arithmetic are exten*e* to ne3ati)enumbers"
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
25/64
◦ 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
$ri3in
: units
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
26/64
0.ample◦
& an( 4& are opposites of each other"
◦
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
27/64
0.ample◦ 4) = 494); = )"
Ce$nition 12 Absolute 4alue #*or an% number a,
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
28/64
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"
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
29/64
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 $n( the sum of to numbers ith unliJe signs, subtract theirabsolute +alues"
The sum is positi+e if the number ith the larger absolute +alue is positi+e The sum is negati+e if the number ith the larger absolute +alue is negati+e"
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
30/64
Subtraction
4 ) # 3 = I = 4) H 943; B% Ce$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;
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
31/64
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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
32/64
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
33/64
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$nition 1 Iultiplicati)e 2n)erse Propert-*or an% nonDero real number a, there is a uniue number 1/asuch that
7f the number is negati+e then its reciprocal is also negati+e"
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
34/64
Di)ision
0.ample◦ =
◦ =
◦ = 2
Ce$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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
35/64
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
36/64
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("
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
37/64
0+aluating 0.pressions
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
38/64
7n algebra %ou ill learn to orJ ith +ariables" oe+er, there is oftennothing more important than $n(ing a numerical anser to a uestion
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
39/64
◦ Example >: H C#C/ # in+ol+es more than one operation of arithmetic
>: H C11
>8 9=H 89= H 9#8
>89/18 9 ,he absoulte )alue is a 3roupin3 s-mbol aswell
>/1
Arithmetic Expression # The result of riting numbers in a meaningfulcombination ith the or(inar% operations of arithmetic
C ! Parentheses are use* as 3roupin3 s-mbols to in*icate whichoperations are performe* rst+ K L ! Brackets are also use* to
in*icate 3roupin3
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
40/64
Exponents ! %otation use* to simplif- the writin3 of a repeate*multiplication
Ce$nition 2' Exponential Expression
*or an% natural number n an( real number a,
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
41/64
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*+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
42/64
Example
0+aluate 91' # ; an( 954; $rst
Then (i+i(e 2 b% 42
91
7hen an expression in)ol)es a fraction bar@ the numerator an**enominator are each treate* as if the- are in parentheses+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
43/64
Ce$nition 22 $r*er of $perations0+aluate insi(e an% grouping s%mbols $rst+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
44/64
Properties of 8eal !umbers
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
45/64
Example : H / > / H :
? > ?
Ce$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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
46/64
Example
C H : H ? > H C: H ? C H ? > H C1/
1= > 1=
Ce$nition 2 Associati)e Propert- of A**ition*or an% real numbers a, b an( c,
CaHb Hc > a H Cb H c+
Associati)e Propert- of Iultiplication*or an% real numbers a, b an( c,
9ab;c = a9bc;"
#C/C: >C#C/C:#C1: > /0
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
47/64
/Cx9# > /Cx H C9# Denition of Subtraction of eal%umbers
> /x H C9 /x 9 ab H ac+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
48/64
Ce$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+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
49/64
Ce$nition 2)A**iti)e 2n)erse Propert-
*or an% real number a, there is a uniue number # a suchthat
aHC9a > 9a H a > 0+
Iultiplicati)e 2*entit- Propert-*or an% real number a, there is a uniue number 91/a; such that
a C16a > 16a Ca > 1+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
50/64
Ce$nition 2 Iultiplication Propert- of Jero*or an% real number a,
0 Ca > a C0 > 0+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
51/64
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@ #@ /@ @ :@
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
52/64
Exercises◦ 5sin3 the sets A@ B@ (@ an* %+ Determine
whether each statement is true or false+
◦ A > &1@ /@ :@ =@ '
◦ B > @ @ &1@ #@ /@ +++' 1 5 B
# B
/ :=!
!=A
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
53/64
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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
54/64
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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
55/64
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
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
56/64
Exercises◦ .ist the elements in each set an* 3raph
each set on a number line+
1 &x8 x is a whole number smaller than
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
57/64
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 /+
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
58/64
Exercises◦ E)aluate+
1 89/8
# 808 / 89
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
59/64
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
60/64
Exercises◦ E)aluate+
1 = ! 10 >
# 91 9 : > / 9 ! < >
#0 ! C9/ >
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
61/64
Exercises◦ E)aluate+
1 #:C9/ >
# C:C9= > / :C: >
9< C9 >
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
62/64
Exercises◦ in* the multiplicati)e in)erse of each
number
1 #0 > # 9: >
/ 9
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
63/64
Exercises◦ E)aluate+ 2f a ;uotient is un*ene*@ sa- so+
1 >
# > / >
-
8/19/2019 Sets Real Numbers and Operations on Real Numbers
64/64