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Press RETURN to continue or press HELP$U* question $1&$U* ask $17$U* ex" "$1G$Udef" "$1Y$Uexplain $1e$U* ovz$U* directory *$Uend$Uundo$Urev" "$1$Ufeedback$Uhow am i doing*%$U* go to $1:TST: $1:EX: $1:DEF: $1OVERVIEWQUITRESUME$3:REV: $1SCORE:GO: $1$U:*:* .sts.%$U:*:* .vnn.7$U:*:* .rst.I$U:*:* .prt.[$U:*:* .sel.m$U:*:* .eps.$U:*:* .fin.$U:*:* .inf.$U:*:* .emp.$U:*:* .eqi.$U:*:* .equ.$U:go:*.sts.%$U:go:*.vnn.7$U:go:*.rst.I$U:go:*.prt.[$U:go:*.sel.m$U:go:*.eps.$U:go:*.fin.$U:go:*.inf.$U:go:*.emp.$U:go:*.eqi.$U:go:*.equ. $U:tst:*.sts.'$U:tst:*.vnn.:$U:tst:*.rst.M$U:tst:*.prt.`$U:tst:*.sel.s $U:tst:*.eps.$U:tst:*.fin. $U:tst:*.inf.+$U:tst:*.emp.6$U:tst:*.eqi.A$U:tst:*.equ.,:#Q'f})  !$U$1" it"$U$1" IT"0$U$1" it "*A$U$1" IT "*T$U*definitionc$U*defines$U*example$U*review$U*quiz$U*test$U*goto$U*what$U*show$U*teach$U*definition" "$1$U*define" "$1$U*example" "$1&$U*review" "$18$U*quiz" "$1J$U*test" "$1\$U*goto" "$1n$U*what" "$1$U*show" "$1$U*teach" "$1$1 .it.:def: .it.:ex: .it.$3:rev: .it.:go: .it.:tst: .it.:def: $1:ex: $1$3:rev: $1:go: $1:tst: $1$U*score*"$U*overview*0$Uresume>$UrepeatJ$UquitV$Uexitd$U*help*n$U?*$U:rev: all$U:rev: all" "* $U:tst: all $U:tst: everything$Urestart$U*index*$U*again*$U:*: *- .$3/ Sets and Notation Disk 1:$U:DEF:*.sts.&=$U:EX:*.sts.9@$U:REV:*.sts.$U$1.it.$1 set'($U* next *:go: .vnn.Sets+2#AS%>#AS%#Press RETURN to continue or press HELP(Press RETURN to continue or press HELP(Introduction to setsDefinition of setsPress RETURN to continue or press HELPExample of setExample of setExample of set Review of sets Test of sets $U$3($Ubald:$U*" bald "*L$U*" BALD "*\$Ubald" "*l$U*" bald"|$U*" BALD"#AS% !  Test of sets $U$3($Ubald:$U*" bald "*L$U*" BALD "*\$Ubald" "*l$U*" bald"|$U*" BALD"#AS% "Review of setsTest of sets$U$3%P$U3/P$U3)<P$UthreeMP$U* pencil*[$#AS%%& Test of sets$U$3%P$U3/P$U3)<P$UthreeMP$U* pencil*[$#AS%%'Review of setsExample of sets=$U:DEF:*.vnn.'@$U:REV:*.vnn.9:$U:EX:*.vnn.-2/!$U$1.it.$1 venn diagrams12$U* next *:go: .rst.*Venn diagrams01245.5#AV%>#AV%7*5#AV%>#AV%Press RETURN to continue or press HELP.34Press RETURN to continue or press HELP246Introduction to Venn diagramsDefinition of Venn diagramPress RETURN to continue or press HELPExample of Venn diagramExample of Venn diagramTest of Venn diagrams%%$U$3.$Uy9$UyesJ$U*" yes "*[$U*" YES "*j$Uyes" "*w$Uy" "*$Uyes," "*$U*" yes"$U*" y"%6$#AV%Review of Venn diagramsExplanation of set notation:$U:DEF:*.rst.&=$U:EX:*.rst.9@$U:REV:*.rst.;<>$U$1.it.$1 roster form/0$U* next *:go: .prt.9Roster form?@ABCDE29#AR%>#AR%KPress RETURN to continue or press HELP@APress RETURN to continue or press HELP=BCDPress RETURN to continue or press HELP@ABCDHDefinition of roster formDefinition of roster formExample of roster form Example of roster form!Example of roster form"Test of roster form###$U$3/O$U*2 *;O$U*6 *RZ$U*{ 1 , 3 , 5 }*iZ$U*{ 1 , 5 , 3 }*Z$U*{ 3 , 1 , 5 }*Z$U*{ 3 , 5 , 1 }*Z$U*{ 5 , 1 , 3 }*Z$U*{ 5 , 3 , 1 }*h$U*{ 1 ; 3 ; 5 }*h$U*{ 1 ; 5 ; 3 }* h$U*{ 3 ; 1 ; 5 }*!h$U*{ 3 ; 5 ; 1 }*8h$U*{ 5 ; 1 ; 3 }*Oh$U*{ 5 ; 3 ; 1 }*%HI$#AR%F$G#AR%Roster form$Roster form%Review of roster formPress RETURN to continue or press HELP&Test of roster form###$U$3/O$U*2 *;O$U*6 *RW$U*{ 1 , 3 , 5 }*iW$U*{ 1 , 5 , 3 }*W$U*{ 3 , 1 , 5 }*W$U*{ 3 , 5 , 1 }*W$U*{ 5 , 1 , 3 }*W$U*{ 5 , 3 , 1 }*e$U*{ 1 ; 3 ; 5 }*e$U*{ 1 ; 5 ; 3 }* e$U*{ 3 ; 1 ; 5 }*!e$U*{ 3 ; 5 ; 1 }*8e$U*{ 5 ; 1 ; 3 }*Oe$U*{ 5 ; 3 ; 1 }*%J$F#AR%$G#AR%Roster form'Test of roster form(##$U$3,$Uc6$Uc)U$U*{ 20 ; 22 ; 24 ; 26 }*t$U*{ 20 , 22 , 24 , 26 }*%LN$#AR%Review of roster formPress RETURN to continue or press HELP&Roster form)Test of roster form(##$U$3,z$Uc6z$Uc)Tz$U*{20 ; 22 ; 24 ; 26 }*rz$U*{20 , 22 , 24 , 26 }*%M$#AR%:$U:DEF:*.prt.&=$U:EX:*.prt.9@$U:REV:*.prt.WRT#$U$1.it.$1 partial listing34$U* next *:go: .sel.PPartial listingUVWXYZ[6=#AP%>#AP%_Press RETURN to continue or press HELPSXYZPress RETURN to continue or press HELPWXYZ\`*Introduction to partial listing+Definition of partial listingPress RETURN to continue or press HELP,Example of partial listing-.Example of partial listing/Example of partial listing01Test of partial listing2''$U$34$U*!{!*A$U*!}!*P$U*!...!*o$U*{ 3 , 5 , 7 , . . . }*$U*{ 3 ; 5 ; 7 ; . . . }*%\]$#AP%Review of partial listing3456Test of partial listing2''$U$34$U*!{!*A$U*!}!*P$U*!...!*o$U*{ 3 , 5 , 7 , . . . }*$U*{ 3 ; 5 ; 7 ; . . . }*%^$#AP%Partial listing7Test of partial listing8''$U$34$U*!{!*A$U*!}!*P$U*!...!*_$U*!100!*$U*{ 1 ; 2 ; 3 ; . . . ; 100 }*$U*{ 1 , 2 , 3 , . . . , 100 }*%`a$#AP%Review of partial listing9:;<=Test of partial listing8''$U$34$U*!{!*A$U*!}!*P$U*!...!*_$U*!100!*$U*{ 1 ; 2 ; 3 ; . . . ; 100 }*$U*{ 1 , 2 , 3 , . . . , 100 }*%b$#AP%Partial listing>:$U:DEF:*.sel.&=$U:EX:*.sel.9@$U:REV:*.sel.lgi$U$1.it.$1 selector,-$U* next *:go: .eps.dSelector methodkjmlnop6=#AL%>#AL%txyzdp#AL%>#AL%tPress RETURN to continue or press HELPhmnoPress RETURN to continue or press HELPlnorwIntroduction to selector method?@Definition of selector methodPress RETURN to continue or press HELPAExample of selector methodBExample of selector methodCDExample of selector methodEUse cursor control keys or mouseTest of selector methodFII$U$3R]J%rq$#AL%F$U$3J%s$#AL%Review of selector methodPress RETURN to continue or press HELPGSelector methodPress RETURN to continue or press HELPHTest of selector methodI''$U$32$U!{!=$U!}!H$U!x!f$U*{x : x *greater* 5 }*|$U*{x : x > 5 }*$U*{x : x *greater* five }*$U*{x  x *greater* 5}*$U*{x  x *> 5 }*$U*{x  x *greater* five }*%wu$#AL%Test of selector methodI''$U$32$U!{!=$U!}!H$U!x!f$U*{x : x *greater* 5 }*|$U*{x : x > 5 }*$U*{x : x *greater* five }*$U*{x  x *greater* 5}*$U*{x  x *> 5 }*$U*{x  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or press HELPPress RETURN to continue or press HELPbcDefinition of finite setPress RETURN to continue or press HELPdeExample of finite setfgExample of finite sethTest of finite setsi##$U$3.$Uyes7$UyF$Uyes" "*V$Uyes," "*e$U*" yes"v$U*" yes "*$U*" YES "*$U*" y "*$Uy" "*%$#AF%Review of finite setsjReview of finite setskTest of finite setsi##$U$3.$Uyes7$UyF$Uyes" "*V$Uyes," "*e$U*" yes"v$U*" yes "*$U*" YES "*$U*" y "*$Uy" "*%$#AF%Test of finite setsl##$U$3,$Ub6$Ub)C$Ub" "*Q$Ub)" "*$U*{ 1975 , 1976 , 1977 , . . . , 1982 }*%$#AF%Review of finite setskTest of finite setsl##$U$3,$Ub6$Ub)C$Ub" "*Q$Ub)" "*$U*{ 1975 , 1976 , 1977 , . . . , 1982 }*%$#AF%m:$U:DEF:*.inf.&=$U:EX:*.inf.9@$U:REV:*.inf.!$U$1.it.$1 infinite sets12$U* next *:go: .emp.Infinite sets.5#AI%>#AI%Press RETURN to continue or press HELPPress RETURN to continue or press HELPnDefinition of infinite setPress RETURN to continue or press HELPopExample of infinite setqExample of infinite setrTest of infinite setss%%$U$33P$UfiniteEP$Ufinite" "*%$#AI%Review of infinite setstTest of infinite setss%%$U$33M$UfiniteEM$Ufinite" "*%$#AI%Review of infinite setsuTest of infinite setsv%%$U$35T$UinfiniteIT$Uinfinite" "*%$#AI%Review of infinite setswTest of infinite setsv%%$U$35Q$UinfiniteIQ$Uinfinite" "*%$#AI%Infinite setsxTest of infinite setsy%%$U$3.$Uc8$Uc)E$Uc" "*T$U*" c "*a$U*" c"y$U* whole number *%$#AI%Review of infinite setszTest of infinite setsy%%$U$3.$Uc8$Uc)E$Uc" "*T$U*" c "*a$U*" c"y$U* whole number *%$#AI%Infinite sets{:$U:DEF:*.emp.&=$U:EX:*.emp.9@$U:REV:*.emp.±$U$1.it.$1 empty)*$U* next *:go: .eqi.Empty sets±ñıű.5#AM%>#AM%Press RETURN to continue or press HELPPress RETURN to continue or press HELP±ñıűDZ|Definition of empty setPress RETURN to continue or press HELP}Example of empty set~Example of empty setExample of empty setTest of empty sets""$U$3-$U{ }6$U0G$U*" { } "*V$U*" 0 "*e$U*" { }"r$U*" 0"$U{ }" "*$U0" "*%$#AM%Review of empty setsTest of empty sets""$U$3,$Uno5$UnD$U*" n "*S$U*" N "*a$U*" no"o$U*" NO"}$Uno" "*%$#AM%Review of empty sets:$U:DEF:*.eqi.&=$U:EX:*.eqi.9@$U:REV:*.eqi.ѱͱ$U$1.it.$1 equivalent./$U* next *:go: .equ.Equivalent setsбѱұӱ07#AQ%>#AQ%ֱPress RETURN to continue or press HELPPress RETURN to continue or press HELPѱұӱDefinition of equivalent setsPress RETURN to continue or press HELPExample of equivalent setsExample of equivalent setsTest of equivalent sets''$U$32$Uyes;$UyL$U*" yes "*]$U*" YES "*l$U*" y "*{$Uyes" "*$Uy " "*$U*" yes"$U*" y"%$#AQ%Review of equivalent setsTest of equivalent sets''$U$30$U4<$UfourZ$U*finite and equivalentx$U*equivalent and finite%ٱ$#AQ%Test of equivalent sets''$U$30$U4<$UfourZ$U*finite and equivalentx$U*equivalent and finite%$#AQ%Equivalent setsReview of equivalent setsTest of equivalent sets''$U$30r$Ub:r$Ub)gr$U*{ 2 , 4 , 6 , 8 }*{ b , d , f , g }*%ݱ$#AQ%Test of equivalent sets''$U$30o$Ub:o$Ub)go$U*{ 2 , 4 , 6 , 8 }*{ b , d , f , g }*%$#AQ%Review of equivalent setsReview of equivalent sets:$U:DEF:*.equ.&=$U:EX:*.equ.9@$U:REV:*.equ.$U$1.it.$1 equal sets./$U* next *:go: .sts.Equal sets.5#AU%>#AU%Press RETURN to continue or press HELPPress RETURN to continue or press HELPDefinition of equal setsPress RETURN to continue or press HELPExample of equal setsExample of equal setsExample of equal setsTest of equal sets""$U$3,@$Uno5@$Un%$#AU%Test of equal sets""$U$3,=$Uno5=$Un%$#AU%Review of equal setsEqual setsUse cursor control keys or mouseTest of equal setsDD$U$3MXJ%$#AU%Use cursor control keys or mouseTest of equal setsDD$U$3MUJ%$#AU%Equal setsPress RETURN to continue or press HELPPress RETURN to continue or press HELPReview of equal setsTest of equal sets""$U$3-$Uyes6$UyG$U* "yes" *X$U* "YES" *g$Uyes" "*w$Uyes," "*$Uy " "*$U*" yes"$U*" YES"%$#AU%Review of equal setsSets and NotationType your answer, press RETURNPress RETURN to continue or press HELP**=*S*h*******.#AS#AV#AR#AP#AL#AE#AF#AI#AM#AQ#AU )8Oc{ʴ޴  Press RETURN to continue or press HELPPress RETURN to continue or press HELPPress RETURN for DESKTOPPress RETURN to continue Review/>TiϱPress RETURN to continue or press HELP#AS%>#AS%#5)0#AV%>#AV%E?F#AR%>#AR%K[X_#AP%>#AP%_pqx#AL%>#AL%t#AE%>#AE%#AF%>#AF%#AI%>#AI%#AM%>#AM%ȱ#AQ%>#AQ%ֱڱ#AU%>#AU%Sets and Notation Disk 1Press RETURN to continue or press HELPSets and Notation Disk 15D#AS%>AR#AS%>c$7masteryq$7needs workq$7incomplete}#AV%>#AV%>$2mastery$2needs work$2incomplete#AR%>#AR%>$3mastery$3needs work$3incomplete #AP%>*#AP%>;$4masteryI$4needs workI$4incompleteUd#AL%>ar#AL%>$5mastery$5needs work$5incomplete#AE%>#AE%>$6mastery$6needs work$6incomplete$3Sets and Notation Disk 15D#AF%>AR#AF%>c$7masteryq$7needs workq$7incomplete}#AI%>#AI%>$5mastery$5needs work$5incomplete#AM%>#AM%>$6mastery$6needs work$6incomplete #AQ%>*#AQ%>;$3masteryI$3needs workI$3incompleteUd#AU%>ar#AU%>$4mastery$4needs work$4incomplete$3You have completed(DISK( 1.8You can review the material on this disk or proceed toXInsert the second disk and "reboot"PdiskP 2.hthemachine.USE THESE CONTROL.intersectionControl T%epsilon Control E7not element Control N@not subsetControl SIsubsetControl BRsupersetControl Pdunion of setsControl U[ "such that"Control V CHARACTERS FOR: The concepts coveredin this tutorial are presented in a predetermined order. (You may, however, interrupt the program AT ANY TIME to: @* ask questions * test your progress * review a concept * advance to a new concept qquestion and press qRETURN. hJust type your message orUSE THESE WORDS TO:*Go to another concept:goto, show me, teach(*Review a concept:0review, review all@*Test your progress:HexHam, testX*Ask questions:define,`example, what isp*Other commands:pscorep, overview,xquit, xagain, restart, resume@SETS In everyday life, we often use terms such as herd, heap, class, group, family, and collection.PThese terms haveXalmost the same meaning as SET.0A set is:@* any collection ofHobjects0A set is:@* any collection ofHobjectsX* clearly described or`identified The objects in the set are * ELEMENTS(or2NUMBERSP* BELONG TO @and are said toXorb* BE CONTAINED IN pthe set. `This is a set of cyclists onphi-rise bikes.This is a set of cyclists on racing bikes.XA collection of records can constitute a SET of records`. Given an object, youhcan tell immediately whether or not it is a record.Since we cannot agree on the meaning of words such as: - honest - sincere - beautiful - funny - old - stupidXwords like these cannot be used to describe a set.p\$3Which one of the following words makes this man a member of a set?@GreatPBald`HonestpStupidPThis man is a member of the set of BALD people.p\$3Which one of the following collections constitutes a set?01) A collection of happy people@2) A collection of good menP3) A collection of pencils`4) A collection of interesting hbooks0The collection of pencils@can constitute a set.p\$30The collection of all schools is a SET since, given an institution we can tell immediately whether or not that institution is a school. @VENN DIAGRAMS 0There are many ways to describe sets. EOne way is to use Venn diagrams.A VENN DIAGRAM is:* a drawing that is used to isolate the elements of a setA VENN DIAGRAM is:* a drawing that is used to isolate the elements of a set8* usually but @not necessarily drawn in the form of a circleA VENN DIAGRAM is:* a drawing that is used to isolate the elements of a set8* usually but @not necessarily drawn in the form of a circle`* used to show thehrelationship between sets0To describe a set and isolate its elements, you can draw a Venn diagram. To describe the set whose elements are the numbers 1, 2, and 3, we can draw this Venn diagram.Is this a Venn diagram? HRemember: XA Venn diagram can be ANY shape that encloses the elements of a set.t\$3 tAll these are Venn diagrams. In addition to  Venn diagrams,(sets can be described using these other forms of written HRoster formWPartial listingfSelector method8notation:@ROSTER FORM(ROSTER FORM is:8* a notation used to@describe a set by listingALLH H itsPelements.In ROSTER FORM,(the elements of the set are:8* separated by commas orDsemicolonsT* enclosed in braces or curlyabrackets, which look likenthis "{}"In roster form, the set whose only elements are the numbers 1, 2, and 3 is written like this:8{1, 2, 3} or {1; 2; 3}.HIt is read, X"the set whose elements are 1, 2, and 3."lNote the commas and semicolons.You can label a set using any0For example:@N = { 1,2,3 } capital letter.PThis is read,`"N is the set whose elements are 1, 2, and 3."To remember the roster form of representing sets, just think of a rooster.@{{{P{{{`The curly brackets are like the rooster's footprints.8Using roster form, describe the following set:P"The set whose elements are 1, 3, and 5."0It is also correct to use semicolons, like this:H{1;3;5}p\$30It is also correct to use commas, like this:H{ 1, 3, 5}p\$3Remember: In set notation, curly brackets are used to enclose the elements`are used to separate the elements.Xcommas or semicolons Hand8of the setp\$3Using roster form, "the set/whose elements are 1, 3, and 5"?can be described as:W{ 1, 3, 5 }or{ 1; 3; 5 }p\$3 Choose the correct use of roster form. 8a)(2O, 22, 24, 26) Pb){2O: 22: 24: 26} hc){2O; 22; 24; 26}The correct use of roster form is:4{2O,22,24,26} or {2O;22;24;26}.GThe elements must be separatedWby commas or semi-colons andgenclosed with curly brackets.p\$3 @PARTIAL LISTING(Sometimes8so many 0a set has @elements that listing them all would be impractical.pIn such a case, a PARTIAL LISTING is used.In partial listing,only the first(few elements are listed.8They are:H* separated by commas or semicolonsX* followed by three dots, which represent the other elementsthree elements of a setare shown.In a partial listing usually8For example: N = {1, 2, 3,...}three elements of a setare shown.8For example: N = {1, 2, 3,...}HWhen the final element of a set is known, it is shown after the three dots.hFor example: N = {2, 4, 6,...,2O}In a partial listing usually The partial listing of all odd numbers after 24 may be written: 8N = {25, 27, 29,...}HThis is read:X"N is the set whose elements are 25, 27, 29, and so on indefinitely."To describe the alphabet using partial listing, we write: 0{a, b, c,...,z}HThis is read, "the set whose elements are the letters a, b, c, and so onto z."To describe the alphabet using partial listing, we write: 0{a, b, c,...,z}HThis is read, "the set whose elements are the letters a, b, c, and so ontohNotice the comma before and after the8Xz."pthree dots.8Describe the following using partial listing:P"The set of odd numbers greater than 2."Remember:The form of a partial listing when the final element is not known 0* the label(includes:p\$3Remember:The form of a partial listing when the final element is not known 0* the curly brackets(includes:p\$3Remember:The form of a partial listing when the final element is not known 0* three elements h* commas or semicolonsp\$3 pbetween the elements(includes:Remember:The form of a partial listing when the final element is not known 0* the three dots(includes:p\$3 The correct way to describe the set of odd numbers greater than 2 in partial listing is:@A @ = { 3, 5, 7,... }orPA P = { 3; 5; 7;... }p\$38Describe, using partial listing, the set whose elements are 1, 2, 3, and so on to 1OO.Remember:The form of a partial listing when the final element is known includes: 0* the labelx\$3Remember:The form of a partial listing when the final element is known includes: 0* the label h* the curly bracketsx\$3Remember:The form of a partial listing when the final element is known includes: 0* three elements h* commas or semicolons between pthe elementsx\$3Remember:The form of a partial listing when the final element is known includes: 0* three dots h* a comma or semicolon after pthe three dotsx\$3Remember:The form of a partial listing when the final element is known includes: 0* the final elementx\$3&The correct way to describe the set of whole numbers from 1 to61OO 6in partial listing is:FA= F{ 1, 2, 3,..., 1OO }orVA = V{ 1; 2; 3;...; 1OO }p\$30A formula is a 0fixed form@of @words or signs @used toPdescribe a rule or idea. @SELECTOR METHOD SELECTOR METHOD0describes a set using:@* a formula or a ruleP Hwithin brackets "{}"0Some sets can be described by using a rule or a formula. In such cases, the SELECTOR METHOD is used. in SELECTOR METHOD, we write: 0B = {x0x is a bicycle}To represent the set of bicycles in SELECTOR METHOD , we write:HThis is read:X"B is the set of all x such that x is a bicycle." 0B = {x0x is a bicycle}To represent the set of bicycles To describe the set "B" of all positive numbers less than 7 inselector method (SET-BUILDER NOTATION), write: PB={xPxP isP a positiveXnumber less than 7}.Point to the correct use of the selector method to describe this set:0A = {a,b,c,...,z}PA = {xx is a,b,c, or z}`A = {xx is a letter}pA = {xx is all letters} )Remember: 9In selector method, x represents one element of the set.p\$3(Using the selector method,8A = {x8]8x is a letter} Hbest describes the setXA = {a,b,c,...,z}.p\$3 ,Using the symbol ,x, 4describe the set whose elements are numbers greater than 5.hUse Control h V for h, the set-builder  In selector method,0"the set of all x such that x is@greater than 5" is written asPfollows: `A = {xx > 5}p\$3Remember: In selector method, a set is represented using:8* curly brackets "{}"L* the set builder "L" or ":"`* the rule (the descriptionhof an element of the set)p\$3*You have now seenhow*to2describe a set by using: BVenn diagrams Roster form Partial listing Selector method.!Try to complete the following exercise on paper.9Using each of the four set notations, describe the set whose elements are:Y1,2,3,4,5,6,7,8,9, and 1O.pSelector Method PF {,1O}PRoster orm1,2,3,4,5,6,7,8,9`Partial Listing {1,2,3,...,1O}x{xxx is a number from 1 to 1O} Venn Diagram @EPSILON SYMBOL * Epsilon is a Greek letter (written like this:8E * Epsilon is a Greek letter (written like this:8E H* It is used to indicate Pthat an element belongs to a set.8In general,Hif "a" belongs to the set X, we write:[aX[E8In general,Hif "a" belongs to the set X, we write:[a [E [Xpelement (member) of set X."hThis is read, "a is anUse the epsilon symbol to indicate that "2" belongs to the set A = {2,4,6,8} by writing:H2 HHAUse the epsilon symbol to indicate that "2" belongs to the set A = {2,4,6,8} by writing:H2 HHA`This is read, "2 is an element (member) of A."Sunday is a member of the set D = {x x is a day of the week}.8This can be written using epsilonHas follows:`Sunday `E` DTo indicate that an element does NOT belong to a set, a bar is placed through the epsilon, like this: We use N  to indicate that the month(of June does NOT belong to the set D = {xD = {x 0x is a day of the week}9by writing:HJune HN HD We use N  to indicate that the month(of June does NOT belong to the set D = {xD = {x 0x is a day of the week},9by writing:HJune HNH DXThis is read: `"June is NOT an element of D." (Which symbol,,(E(or(N 0should replace the question mark: H9 ? B = {2,4,6,...} XUse ControlXN for XNhControlh E for hRemember:(The symbol(N(means0"is not an element of."@For example, P9 PN P B = {2,4,6,...}`because 9 is an odd number.p\$3 If A={f,j,m} and B={s,w,z},-which statement is correct?E1)mENEAU2U) UsU3UEUBe3)f eB Remember:E."means "is an element of,For example, <3 <E < B = {1,2,3,4,5}Lwhich is read,\"3 is an element of set B,dwhose elements are the numbers 1 through 5."t\$3 If,0A = {f,j,m} and B = {s,w,z}@then the statement "s is anPelement of B" is correct. `s `` B q\$3+IfA={x hUse Control E for pControl N for +x is a whole number3from 4 to 9},;which symbol, ;should;E;or;N,Creplace the blank?S4AT_ A = {xx is a whole number(from 4 to 9}8Since set A includes the Hnumbers 4,5,6,7,8, and 9,X4 is a member of the set.h4  Aq\$3 We will now look at the following different types@FINITE SETS INFINITE SETS EMPTY SETS 0of sets:@FINITE SETS(A (FINITE( set:8*8is a set in which the elements can be completely counted(A(FINITE( set:8*8is a set in which the elements can be completely countedX* contains a limited`number of elements`The set of hairs on your head is a finite set because it is possible to count every hair. `The set of hairs on your head is a finite set because it is possible to count every hair.The set of letters in the English alphabet is a finite set since all the letters can be counted.x p h 8Is the set of astronauts who have landed on the moon a finite set?p\$3(A set is FINITE if its elements can be counted completely.@For example, the set of astronauts who have landed on the moon is FINITE. (Remember: 8A set is FINITE if its Helements can be counted Xcompletely.p\$3Which set, A, B, C or D, is a finite set?8A = {198O, 1981, 1982,...}HB= {1975, 1976, 1977,...,1982}XC= {x Xx is a leap year}hD= {x hx is a year starting from p1975} The set0B = {1975, 1976, 1977,...,1982}@is finite since all the yearsPbetween 1975 and 1982 are`countable.p\$3 @INFINITE SETS An  INFINITE  set is:0* a set whose elements8cannot be counted completely An INFINITE  set is:0* a set whose elements8cannot be counted completelyP* a set that containsXan UNLIMITEDX number`of elementshThe set of numbers {1, 2, 3,...} is an infinite set because it contains an UNLIMITED number of elements.(The sets of even and odd numbers8are both infinite because theirHelements cannot be countedXcompletely.0Is the set of fish in 0L0ake@Huron @ @f @inite or infinite?\$3pAn infinite set is a set whose pelementsx cannot be counted completely.0Since the set of fish in 0Lake@Huron @ can be counted completelyPit constitutes a finite set.p\$38Is the following set finite or infinite?PCP = {25, 5O, 75,...}p\$3@An infinite set is a set whose elements cannot be counted completely. (C = {25,5O,75,...} @Set C is an example Pof an infinite set.p\$3.Which of the following; A, B., or C,7is an example of an infinite set?GA = {x Gx is an intelligent student}WB = {x Wx is a word in a dictionary}gC = {x gx is a whole number}p\$3({x(x is an intelligent student} is0neither infinite nor finite because it is not a set.HNot everybody agrees on the meaning of words such as "intelligent."0 C = { xx is a whole number}@is a good example of an infinitePset.p\$3@EMPTY SETSAn EMPTY set:0contains NO ELEMENTS(* is (a set that@* is also called @a"@ HNULL SETX* is represented by the`symbols 0 `or { }hEither the symbol 0 or the symbol { } is used to represent this set.#( (The set of people who are 2O feet tall is8 an empty, or null@set, because people that tall do not exist.The set of people(who have three legs8is an empty set Hbecause there are Xno people with threehlegs.0Type in one of the symbols@usedtorepresent@an emptyPset.p\$3 8The two symbols used @to represent a null, or empty set are:X 0, or { } 8Is 0 the same as {0}?HYes or no.p\$30 and {} both represent a null or empty set,0and@{0} represents a set containing one element, 0, (the null set).XTherefore, 0 is not the same as {0}. @EQUIVALENT SETS(Equivalent sets are:8* sets that contain@the SAME NUMBER of elements Sets C and A are equivalent sets because they contain the same number of elements.HC H=hA=P,P,p,p,0Sets D and E are equivalent because they contain the SAME NUMBER of elements. D= {7,8,9,1O,11}, E= {a,b,c,d,e}0Sets D and E are equivalent because they contain the SAME NUMBER of elements.PNote that equivalent sets do not need to contain the same elements. D= {7,8,9,1O,11}, E= {a,b,c,d,e}4Are the following sets equivalent?\D= {7,K,22}LC= {A,Z,K}Uelements.#Sets # C= {A,Z,K}#and D= {7,K,22}3are equivalent.p\$3ETEhey contain the SAMEENUMBER of A= {7,8,9,1O}4Choose the most accurate statement to describe sets A and B.L1) A and B are infinite sets 2) A and B are finite sets 3) A andB are equivalent sets 4) A and B are both finiteland equivalent sets B= {5,7,9,13}0The sets,@A = {7,8,9,1O} and B = {5,7,9,13}Pare FINITE and EQUIVALENT.p\$3Remember:(Sets A = {7,8,9,1O}8and B = {5,7,9,13}Hcontain a LIMITED Xand EQUAL number ofhelements.p\$30of sets.Ha) {0} 0(Choose the equivalent pairXb) {2,4,6,8} {b,d,f,g}hc) {2,4,6,...} {2,4,6} {2,4,6,8} {b,d,f,g}8These sets are equivalentHsince they contain theXSAME NUMBER of elements.p\$3p\$38Equivalent sets must contain HtheS HAME NUMBER OF ELEMENTS.@EQUAL SETS E QUAL  sets are sets that: 0* contain exactly the 9 SAME ELEMENTS Q  E QUAL  sets are sets that: 0* contain exactly the 9 SAME ELEMENTS X* contain exactly the a SAME NUMBER OF ELEMENTSJANDNote that all equal sets are also equivalent sets because they contain the same number of elements.HAH H=hB = P,PP,p,pp,Equal sets contain the SAME ELEMENTS and the SAME NUMBER of elements.(Sets A and B are EQUAL.HA H=hB = P,P,p,p,(Let A = {1,2,3} and B = {3,2,1}.>Sets A and B are EQUAL, even though their elements are not listed in the same order. Are these sets equal?kC=ID=P,P,p,,p\$38Equal sets contain both:H- the same elements,X- the same number of elements.Pandp\$3Since the elements in the two sets are%not the same, the sets are equivalent5but not equal.Let A = {3,9,15,2O} and B = {15,2O,9,3}.(Point to the statement0thatB 0EST describ0es8these twosets.@The sets are:P*P equivalent and equalX*X finite and equal`*` equivalent, equal,hand hfinitep*p equivalent and finite "B = {15,2O,9,3}  A = {3,9,15,2O}0These sets are:@* EQUIVALENT because they containHthe same number of elementsP*EQUAL because they contain theXsame elements`* FINITE because they contain a limited number of elementsp\$3 (Given X = {a,e,i,o} 0and Y = {i,a,o,e} @Are these sets equal? 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