Sets are collection of objects that can be displayed in different forms. Two of these forms are called Roster Method and Builder Set Notation.

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1 Sectio 2.1 Set ad Set Operators Defiitio of a set set is a collectio of objects thigs or umbers. Sets are collectio of objects that ca be displayed i differet forms. Two of these forms are called Roster Method ad uilder Set Notatio. Roster Method: I roster method the elemets of the set are listed i brackets ad separated by commos. The sets i the above examples are i roster form. { } { Ro Joh Mark Phil} { Virgiia West Virgiia Marylad Teessee Ketucky Noth Carolia} uilder Set Notatio: I uilder Set Notatio the followig format is used x : x ( descriptio { } Here are some examples of sets that writte i uilder Set Notatio. { x x is a vowel} { x x is a great lake} { x x is a eve atural umber} I order to write a set i uilder Set Notatio you must be able to describe the set. set must be well defied to write i uilder Set Notatio. set is well defied is the elemets of the sets are clearly defied. If a set is well defied the there should ot be ay cofusio of what the elemets are i the set Examples of well defied sets { 135 } { m o p} { x x is a whole umber}

2 Examples of set that are ot well defied { x x is somethig cool} { x x is a small dog} Elemets are the members of a give set. represetsis a elemet of represets is ot a elemet of 3 a { } { a b c d e} asic Number Sets Natural Numbers or Coutig Numbers: N { } Whole Numbers: W { } Itegers I { } Ratioal Numbers: Q {x x is a termiatig umber or repeatig decimal} Irratioal Numbers: J {x x is ot a termiatig umber or repeatig decimal} Real Numbers: R {x x is a ratioal umber or irratioal umber} Practice Problems Example 1 Write the followig set i roster form. The set of the seve dwarfs Solutio: { Dopey Sleepy Grumpy Seezy Happy Droopy Doc } Example 2 Write the followig set i roster form. The set of the five great lakes Solutio: { Huro Otario Michiga Erie Superior}

3 Example 3 Write the followig set i roster form. The set of all itegers { } Example 4 Write the followig set i uilder Set Notatio. { } { x x is multiple of five betwee 10 ad 35} Example 5 Write the followig set i uilder Set Notatio. { Ohio tah Iowa} { x x is a state with four letters} Equivalet Sets Two sets are equivalet if they have the same umber of elemets. Two equivalet sets ad are deoted by ~ Examples of equivalet sets { 1234 } ad { a b c d} { joh luke mark mathew} ad { a b c d}

4 Equal Sets Two sets are equal if their elemets ate idetical. Two equal sets ad are deoted by Example of two equal sets { a b c} ad { c a b} Or { a b c} ~ { c a b} Example 6 Classify as true or false 1 2 { } True 2 is a elemet of the set { } 2 7 { } False 6 is ot i the set { } 3 { 1 35} ~ { a m v} True the two sets have the same umber of elemets. 4 {} 1 { } The elemet{} 1 is ot i the set { }

5 Sectio 2.2 Subsets ad Improper Subsets Key Terms The empty set is a set that cotais o elemets. The empty set is also referred to as the ull set. Subsets set is a subset of set C if every elemet i is a elemet of C. C Proper Subsets set is a proper subset of C if every elemet of is a elemet of C ad there is at least oe elemet of C that is ot i. C Example 1 C Is { } { } C? Solutio: Sice every elemet i the set is a elemet of C is a subset of C. Example 2 Is { 456} a subset of { }? Solutio: o sice the elemet 6 i ot i the set{ } Example 2 Is { 456} a proper subset of { 456}? Solutio: The set{ 4 56} is a subset of itself but ot a proper subset. Remember that the paret set must have at least oe elemet that is ot i the proper subset.

6 Example 4 List all possible subsets of { a m} Solutio: φ {}{ a m}{ a m} Example 5 List all subsets of the set {234} Possible subsets Solutio: φ {}{}{}{ }{ 34}{ 24}{ 234} Example 6 List all subsets of the set {6} Possible sets: φ {} 6 The patter for subsets Number of Number of subsets elemets Formula to fid the umber of subsets s of a give set with elemets s 2

7 Example 7 How may subsets does a set with 10 elemets have? s 2 s 2 10 s 1024 The uiversal set is the set of all possible elemets of set used i the problem. Deoted by The complemet of a set The complemet of a set is the set of all elemets i the uiversal that are ot elemets of the set. { x x ad x } Example 8 Fid the complimet of each set. The that the uiversal set is { } 1 { 2345} { } 2 The odd atural umbers less tha 10: { } Complimet { } 3 { } Complimet { }

8 Sectio 2.3 Set Operators io ad Itersectio io of Two Sets The uio of two sets is deoted by is { x x or x } Itersectio of Two Sets The itersect of two sets is deoted by is { x x ad x } Example 1 Let { } { 1357 } C { 12} { 12} D ad E φ 1 Is C? swer: Yes every elemet i C is cotaied i 2 Is φ? Yes the empty set is a subset of ay oempty every set. 3 Fid swer: { 135 } 4 Fid swer: { } 5 Fid C C swer: { 12} 6 Fid ( C ( C swer: { } ({ } { } { } {} { 1} 7 Fid ( C swer: ( C { } ({ } { } { } { } { }

9 Example 2 Let { a b c d} { a b d e} C { b c d} ad D { c d} 8 Is C? swer: Yes every elemet i C is cotaied i 9 Is φ? Yes the empty set is a subset of ay oempty every set. 10 Fid a b d swer: { } 11 Fid a b c d e swer: { } 12 Fid ( C swer: ( C a b c d a b d e b c d a b c d b d { } ({ } { } { } { } { b d} Ve Diagrams Geeral Ve Diagram for sets ad the uiversal set The Ve diagram for The Ve diagram for

10 The complemet of a set The complemet of a set is the set of all elemets i the uiversal that are ot elemets of the set. { x x ad x } Example 3 1 Fid 567 { }

11 2 Fid Fid 8910 { } { } Example 3 Give Fid ( { } { 45678} { } 2 { } { 456} { } { } 5 ( ({ } { 45678} { } { 456} { } { } { } { }

12 6 Make a Ve diagram of ad Ve diagrams Shade the regio correspodig to the idicated set. 1 2

13 3 4

14 Sectio 2.4 pplicatios of Sets Defiitio: ( ( + ( ( Example 1 Give ( 340 ( 240 ad ( 80 fid ( ( ( + ( ( Example 2 Give ( 30 ( 28 ad ( 50 fid ( ( ( + ( ( ( ( 8 ( ( 8 Example 3 Give ( 88 ( 65 ad ( 120 fid ( ( ( + ( ( ( ( 33 ( ( 33

15 Cardiality Defiitio: Cardiality is the umber of elemets i a give set The umber of elemets i a set is deoted by ( { } { } { } k j i h g f e d c b a i h g c b a e d c b a j k g h i a b c d e 1 Fid ( 5 ( 2 Fid ( 6 ( 3 Fid ( 8 ( 4 Fid ( 3 ( Rules for the cardiality for the uio of two sets ( ( ( ( + se this formula to fid ( i problem ( ( ( ( + +

16 Example 5 Let { x x is a state i the ited States } { x x ad x begis with } I { x x ad x begis with I} M { x x ad x begis with M } N { x x ad x begis with N} O { x x ad x begis with O} I M { labama rkasas laska rizoa} { Iowa Idiaa Illiois Idaho} { Michiga Miesota Mississippi Missouri Marylad Maie Mo ta a Massachusetts} Nebraska New Jersey New Mexico New York New Hamphere North Carolia N North Dakota Nevada O { Ohio Okla hom a Orego } 33 Fid 34 Fid 35 Fid 36 Fid ( M ( N 13 ( I O 50 ( ( M I 0 Example 4 Let { Eglish Math History Drama Physics Spaish Philosohpy Chemistry Lati Frech } { Eglish History Chemistry Spaish} { History Math Chemistry Frech} C { Physics Eglish Frech Math} 1 Fid { Eglish History Chemistry Spaish Math Frech} 2 Fid { Eglish Chemistry}

17 3 Fid ( ( 6 4 Fid ( ( 2 5 Fid ( + ( ( + ( Fid ( + ( + ( C ( + ( + ( C Sectio 2.5

18 Ifiite Sets Ifiite sets ad Cardiality Equivalet Sets Two sets are equivalet if they have the same umber of elemets. Examples of equivalet sets { 1234 } ad { a b c d} { joh luke mark mathew} ad { a b c d} Cardiality Defiitio: Cardiality is the umber of elemets i a give set Oe-to-oe correspodece Defiitio: Two sets are i oe-to-oe correspodece if each elemet i the first is paired with exactly oe elemet i the secod set ad each elemet of the secod set is paired with exactly oe elemet form the first set Examples 1 The sets {234} ad { a b c d} 1 are i oe-to-oe correspodece as show i this diagram. { } { a b c d}

19 2 The sets{ Virigiia Marylad North Carolia} ad { Richmod apolis Raliegh} are i oe-to-oe correspodece as show i this diagram. { Virigiia Marylad North Carolia} { Richmod apolis Raliegh} Cator s defiitio of set set is ifiite if we ca remove some of its elemets without reducig its size. Coutable sets set is coutable if you establish a oe-to-oe correspodece form the give set to the atural umbers. Examples 1 re the eve atural umbers coutable? { } { } The eve atural ca be put i a oe-to-oe correspodece with the atural umbers by usig the mappig 2

20 2 re the itegers coutable? J { } The mappig would go as follows: etc. se this mappig if is eve 2 1 if is odd 2 Therefore there exist a oe-to-oe correspodece betwee the itegers ad the atural umbers. Thus the itegers are coutable.

21 3 re the ratioal umbers coutable? Look at the followig diagram This allow the followig orderig of umbers This shows that each elemet of the ratioal umber ca be paired with oe elemet of the atural umbers. Thus it is possible to establish a oe-to-oe correspodece with the atural umbers. This provides a iterestig result which is that the ratioal umbers tur out to coutable.