0.1 Bsic Set Theory nd Intervl Nottion 3 0.1 Bsic Set Theory nd Intervl Nottion 0.1.1 Some Bsic Set Theory Notions Like ll good Mth ooks, we egin with definition. Definition 0.1. A set is well-defined collection of ojects which re clled the elements of the set. Here, well-defined mens tht it is possile to determine if something elongs to the collection or not, without prejudice. The collection of letters tht mke up the word smolko is well-defined nd is set, ut the collection of the worst Mth techers in the world is not well-defined nd therefore is not set. 1 In generl, there re three wys to descrie sets nd those methods re listed elow. Wys to Descrie Sets 1. The Verl Method: Use sentence to define the set. 2. The Roster Method: Begin with left rce {, list ech element of the set only once nd then end with right rce }. 3. The Set-Builder Method: A comintion of the verl nd roster methods using dummy vrile such s x. For exmple, let S e the set descried verlly s the set of letters tht mke up the word smolko. A roster description of S is {s, m, o, l, k}. Note tht we listed o only once, even though it ppers twice in the word smolko. Also, the order of the elements doesn t mtter, so {k, l, m, o, s} is lso roster description of S. Moving right long, set-uilder description of S is: {x x is letter in the word smolko }. The wy to red this is The set of elements x such tht x is letter in the word smolko. In ech of the ove cses, we my use the fmilir equls sign = nd write S = {s, m, o, l, k} or S = {x x is letter in the word smolko }. Notice tht m is in S ut mny other letters, such s q, re not in S. We express these ides of set inclusion nd exclusion mthemticlly using the symols m 2 S (red m is in S ) nd q /2 S (red q is not in S ). More precisely, we hve the following. Definition 0.2. Let A e set. If x is n element of A then we write x 2 A which is red x is in A. If x is not n element of A then we write x /2 A which is red x is not in A. Now let s consider the set C = {x x is consonnt in the word smolko }. A roster description of C is C = {s, m, l, k}. Note tht y construction, every element of C is lso in S. We express 1 For more thought-provoking exmple, consider the collection of ll things tht do not contin themselves - this leds to the fmous Russell s Prdox.
4 Prerequisites this reltionship y stting tht the set C is suset of the set S, which is written in symols s C S. The more forml definition is given elow. Definition 0.3. Given sets A nd B, we sy tht the set A is suset of the set B nd write A B if every element in A is lso n element of B. Note tht in our exmple ove C S, ut not vice-vers, since o 2 S ut o /2 C. Additionlly, the set of vowels V = {, e, i, o, u}, while it does hve n element in common with S, is not suset of S. (As n dded note, S is not suset of V, either.) We could, however, uild set which contins oth S nd V s susets y gthering ll of the elements in oth S nd V together into single set, sy U = {s, m, o, l, k,, e, i, u}. Then S U nd V U. The set U we hve uilt is clled the union of the sets S nd V nd is denoted S [ V. Furthermore, S nd V ren t completely different sets since they oth contin the letter o. The intersection of two sets is the set of elements (if ny) the two sets hve in common. In this cse, the intersection of S nd V is {o}, written S \ V = {o}. We formlize these ides elow. Definition 0.4. Suppose A nd B re sets. The intersection of A nd B is A \ B = {x x 2 A nd x 2 B} The union of A nd B is A [ B = {x x 2 A or x 2 B (or oth)} The key words in Definition 0.4 to focus on re the conjunctions: intersection corresponds to nd mening the elements hve to e in oth sets to e in the intersection, wheres union corresponds to or mening the elements hve to e in one set, or the other set (or oth). In other words, to elong to the union of two sets n element must elong to t lest one of them. Returning to the sets C nd V ove, C [ V = {s, m, l, k,, e, i, o, u}. 2 When it comes to their intersection, however, we run into it of nottionl wkwrdness since C nd V hve no elements in common. While we could write C \ V = {}, this sort of thing hppens often enough tht we give the set with no elements nme. Definition 0.5. The Empty Set ; is the set which contins no elements. Tht is, ; = {} = {x x 6= x}. As promised, the empty set is the set contining no elements since no mtter wht x is, x = x. Like the numer 0, the empty set plys vitl role in mthemtics. 3 We introduce it here more s symol of convenience s opposed to contrivnce. 4 Using this new it of nottion, we hve for the sets C nd V ove tht C \ V = ;. A nice wy to visulize reltionships etween sets nd set opertions is to drw Venn Digrm. A Venn Digrm for the sets S, C nd V is drwn t the top of the next pge. 2 Which just so hppens to e the sme set s S [ V. 3 Sdly, the full extent of the empty set s role will not e explored in this text. 4 Actully, the empty set cn e used to generte numers - mthemticins cn crete something from nothing!
0.1 Bsic Set Theory nd Intervl Nottion 5 U C smlk o eiu S V A Venn Digrm for C, S nd V. In the Venn Digrm ove we hve three circles - one for ech of the sets C, S nd V. We visulize the re enclosed y ech of these circles s the elements of ech set. Here, we ve spelled out the elements for definitiveness. Notice tht the circle representing the set C is completely inside the circle representing S. This is geometric wy of showing tht C S. Also, notice tht the circles representing S nd V overlp on the letter o. This common region is how we visulize S \ V. Notice tht since C \ V = ;, the circles which represent C nd V hve no overlp whtsoever. All of these circles lie in rectngle leled U (for universl set). A universl set contins ll of the elements under discussion, so it could lwys e tken s the union of ll of the sets in question, or n even lrger set. In this cse, we could tke U = S [ V or U s the set of letters in the entire lphet. The reder my well wonder if there is n ultimte universl set which contins everything. The short nswer is no nd we refer you once gin to Russell s Prdox. The usul triptych of Venn Digrms indicting generic sets A nd B long with A \ B nd A [ B is given elow. U U U A \ B A [ B A B A B A B Sets A nd B. A \ B is shded. A [ B is shded.
6 Prerequisites 0.1.2 Sets of Rel Numers The plyground for most of this text is the set of Rel Numers. Mny quntities in the rel world cn e quntified using rel numers: the temperture t given time, the revenue generted y selling certin numer of products nd the mximum popultion of Ssqutch which cn inhit prticulr region re just three sic exmples. A succinct, ut nonetheless incomplete 5 definition of rel numer is given elow. Definition 0.6. A rel numer is ny numer which possesses deciml representtion. The set of rel numers is denoted y the chrcter R. Certin susets of the rel numers re worthy of note nd re listed elow. In fct, in more dvnced texts, 6 the rel numers re constructed from some of these susets. Specil Susets of Rel Numers 1. The Nturl Numers: N = {1, 2, 3,...} The periods of ellipsis... here indicte tht the nturl numers contin 1, 2, 3 nd so forth. 2. The Whole Numers: W = {0, 1, 2,...}. 3. The Integers: Z = {..., 3, 2, 1, 0, 1, 2, 3,...} = {0, ±1, ±2, ±3,...}. 4. The Rtionl Numers: Q = 2 Z nd 2 Z. Rtionl numers re the rtios of integers where the denomintor is not zero. It turns out tht nother wy to descrie the rtionl numers is: Q = {x x possesses repeting or terminting deciml representtion} 5. The Irrtionl Numers: P = {x x 2 R ut x /2 Q}. c Tht is, n irrtionl numer is rel numer which isn t rtionl. Sid differently, P = {x x possesses deciml representtion which neither repets nor termintes} The symol ± is red plus or minus nd it is shorthnd nottion which ppers throughout the text. Just rememer tht x = ±3 mens x = 3 or x = 3. See Section 9.2. c Exmples here include numer (See Section 10.1), p 2 nd 0.101001000100001... Note tht every nturl numer is whole numer which, in turn, is n integer. Ech integer is rtionl numer (tke = 1 in the ove definition for Q) nd since every rtionl numer is rel numer 7 the sets N, W, Z, Q, nd R re nested like Mtryoshk dolls. More formlly, these sets form suset chin: N W Z Q R. The reder is encourged to sketch Venn Digrm depicting R nd ll of the susets mentioned ove. It is time for n exmple. 5 Mth pun intended! 6 See, for instnce, Lndu s Foundtions of Anlysis. 7 Thnks to long division!
0.1 Bsic Set Theory nd Intervl Nottion 7 Exmple 0.1.1. 1. Write roster description for P = {2 n n 2 N} nd E = {2n n 2 Z}. 2. Write verl description for S = {x 2 x 2 R}. 3. Let A = { 117, 4 5, 0.202002, 0.202002000200002...}. () Which elements of A re nturl numers? Rtionl numers? Rel numers? () Find A \ W, A \ Z nd A \ P. 4. Wht is nother nme for N [ Q? Wht out Q [ P? Solution. 1. To find roster description for these sets, we need to list their elements. Strting with P = {2 n n 2 N}, we sustitute nturl numer vlues n into the formul 2 n. For n = 1 we get 2 1 = 2, for n = 2 we get 2 2 = 4, for n = 3 we get 2 3 = 8 nd for n = 4 we get 2 4 = 16. Hence P descries the powers of 2, so roster description for P is P = {2, 4, 8, 16,...} where the... indictes the tht pttern continues. 8 Proceeding in the sme wy, we generte elements in E = {2n n 2 Z} y plugging in integer vlues of n into the formul 2n. Strting with n = 0 we otin 2(0) = 0. For n = 1 we get 2(1) = 2, for n = 1 we get 2( 1) = 2 for n = 2, we get 2(2) = 4 nd for n = 2 we get 2( 2) = 4. As n moves through the integers, 2n produces ll of the even integers. 9 A roster description for E is E = {0, ±2, ±4,...}. 2. One wy to verlly descrie S is to sy tht S is the set of ll squres of rel numers. While this isn t incorrect, we d like to tke this opportunity to delve little deeper. 10 Wht mkes the set S = {x 2 x 2 R} little trickier to wrngle thn the sets P or E ove is tht the dummy vrile here, x, runs through ll rel numers. Unlike the nturl numers or the integers, the rel numers cnnot e listed in ny methodicl wy. 11 Nevertheless, we cn select some rel numers, squre them nd get sense of wht kind of numers lie in S. For x = 2, x 2 =( 2) 2 = 4 so 4 is in S, s re 3 2 2 = 9 4 nd (p 117) 2 = 117. Even things like ( ) 2 nd (0.101001000100001...) 2 re in S. So suppose s 2 S. Wht cn e sid out s? We know there is some rel numer x so tht s = x 2. Since x 2 0 for ny rel numer x, we know s 0. This tells us tht everything 8 This isn t the most precise wy to descrie this set - it s lwys dngerous to use... since we ssume tht the pttern is clerly demonstrted nd thus mde evident to the reder. Formuls re more precise ecuse the pttern is cler. 9 This shouldn t e too surprising, since n even integer is defined to e n integer multiple of 2. 10 Think of this s n opportunity to stop nd smell the mthemticl roses. 11 This is nontrivil sttement. Interested reders re directed to discussion of Cntor s Digonl Argument.
8 Prerequisites in S is non-negtive rel numer. 12 This egs the question: re ll of the non-negtive rel numers in S? Suppose n is non-negtive rel numer, tht is, n 0. If n were in S, there would e rel numer x so tht x 2 = n. As you my recll, we cn solve x 2 = n y extrcting squre roots : x = ± p n. Since n 0, p n is rel numer. 13 Moreover, ( p n) 2 = n so n is the squre of rel numer which mens n 2 S. Hence, S is the set of non-negtive rel numers. 3. () The set A contins no nturl numers. 14 Clerly, 4 5 is rtionl numer s is 117 117 (which cn e written s 1 ). It s the lst two numers listed in A, 0.202002 nd 0.202002000200002..., tht wrrnt some discussion. First, recll tht the line over the digits 2002 in 0.202002 (clled the vinculum) indictes tht these digits repet, so it is rtionl numer. 15 As for the numer 0.202002000200002..., the... indictes the pttern of dding n extr 0 followed y 2 is wht defines this rel numer. Despite the fct there is pttern to this deciml, this deciml is not repeting, so it is not rtionl numer - it is, in fct, n irrtionl numer. All of the elements of A re rel numers, since ll of them cn e expressed s decimls (rememer tht 4 5 = 0.8). () The set A \ W = {x x 2 A nd x 2 W} is nother wy of sying we re looking for the set of numers in A which re whole numers. Since A contins no whole numers, A \ W = ;. Similrly, A \ Z is looking for the set of numers in A which re integers. Since 117 is the only integer in A, A \ Z = { 117}. As for the set A \ P, s discussed in prt (), the numer 0.202002000200002... is irrtionl, so A \ P = {0.202002000200002...}. 4. The set N [ Q = {x x 2 N or x 2 Q} is the union of the set of nturl numers with the set of rtionl numers. Since every nturl numer is rtionl numer, N doesn t contriute ny new elements to Q, so N [ Q = Q. 16 For the set Q [ P, we note tht every rel numer is either rtionl or not, hence Q [ P = R, pretty much y the definition of the set P. As you my recll, we often visulize the set of rel numers R s line where ech point on the line corresponds to one nd only one rel numer. Given two different rel numers nd, we write < if is locted to the left of on the numer line, s shown elow. The rel numer line with two numers nd where <. While this notion seems innocuous, it is worth pointing out tht this convention is rooted in two deep properties of rel numers. The first property is tht R is complete. This mens tht there 12 This mens S is suset of the non-negtive rel numers. 13 This is clled the squre root closed property of the non-negtive rel numers. 14 Crl ws tempted to include 0.9 in the set A, ut thought etter of it. See Section 9.2 for detils. 15 So 0.202002 = 0.20200220022002... 16 In fct, nytime A B, A [ B = B nd vice-vers. See the exercises.
0.1 Bsic Set Theory nd Intervl Nottion 9 re no holes or gps in the rel numer line. 17 Another wy to think out this is tht if you choose ny two distinct (different) rel numers, nd look etween them, you ll find solid line segment (or intervl) consisting of infinitely mny rel numers. The next result tells us wht types of numers we cn expect to find. Density Property of Q nd P in R Between ny two distinct rel numers, there is t lest one rtionl numer nd irrtionl numer. It then follows tht etween ny two distinct rel numers there will e infinitely mny rtionl nd irrtionl numers. The root word dense here communictes the ide tht rtionls nd irrtionls re thoroughly mixed into R. The reder is encourged to think out how one would find oth rtionl nd n irrtionl numer etween, sy, 0.9999 nd 1. Once you ve done tht, try doing the sme thing for the numers 0.9 nd 1. ( Try is the opertive word, here. 18 ) The second property R possesses tht lets us view it s line is tht the set is totlly ordered. This mens tht given ny two rel numers nd, either <, > or = which llows us to rrnge the numers from lest (left) to gretest (right). You my hve herd this property given s the Lw of Trichotomy. Lw of Trichotomy If nd re rel numers then exctly one of the following sttements is true: < > = Segments of the rel numer line re clled intervls. They ply huge role not only in this text ut lso in the Clculus curriculum so we need concise wy to descrie them. We strt y exmining few exmples of the intervl nottion ssocited with some specific sets of numers. Set of Rel Numers Intervl Nottion Region on the Rel Numer Line {x 1 pple x < 3} [1, 3) 1 3 {x 1 pple x pple 4} [ 1, 4] 1 4 {x x pple 5} ( 1, 5] 5 {x x > 2} ( 2, 1) 2 As you cn glen from the tle, for intervls with finite endpoints we strt y writing left endpoint, right endpoint. We use squre rckets, [ or ], if the endpoint is included in the intervl. This 17 Als, this intuitive feel for wht it mens to e complete is s good s it gets t this level. Completeness does get much more precise mening lter in courses like Anlysis nd Topology. 18 Agin, see Section 9.2 for detils.
10 Prerequisites corresponds to filled-in or closed dot on the numer line to indicte tht the numer is included in the set. Otherwise, we use prentheses, ( or ) tht correspond to n open circle which indictes tht the endpoint is not prt of the set. If the intervl does not hve finite endpoints, we use the symol 1 to indicte tht the intervl extends indefinitely to the left nd the symol 1 to indicte tht the intervl extends indefinitely to the right. Since infinity is concept, nd not numer, we lwys use prentheses when using these symols in intervl nottion, nd use the pproprite rrow to indicte tht the intervl extends indefinitely in one or oth directions. We summrize ll of the possile cses in one convenient tle elow. 19 Intervl Nottion Let nd e rel numers with <. Set of Rel Numers Intervl Nottion Region on the Rel Numer Line {x < x < } (, ) {x pple x < } [, ) {x < x pple } (, ] {x pple x pple } [, ] {x x < } ( 1, ) {x x pple } ( 1, ] {x x > } (, 1) {x x } [, 1) R ( 1, 1) We close this section with n exmple tht ties together severl concepts presented erlier. Specificlly, we demonstrte how to use intervl nottion long with the concepts of union nd intersection to descrie vriety of sets on the rel numer line. 19 The importnce of understnding intervl nottion in Clculus cnnot e overstted so plese do yourself fvor nd memorize this chrt.
0.1 Bsic Set Theory nd Intervl Nottion 11 Exmple 0.1.2. 1. Express the following sets of numers using intervl nottion. () {x x pple 2 or x 2} () {x x 6= 3} (c) {x x 6= ±3} (d) {x 1 < x pple 3 or x =5} 2. Let A =[ 5, 3) nd B = (1, 1). Find A \ B nd A [ B. Solution. 1. () The est wy to proceed here is to grph the set of numers on the numer line nd glen the nswer from it. The inequlity x pple 2 corresponds to the intervl ( 1, 2] nd the inequlity x 2 corresponds to the intervl [2, 1). The or in {x x pple 2 or x 2} tells us tht we re looking for the union of these two intervls, so our nswer is ( 1, 2] [ [2, 1). 2 2 ( 1, 2] [ [2, 1) () For the set {x x 6= 3}, we shde the entire rel numer line except x = 3, where we leve n open circle. This divides the rel numer line into two intervls, ( 1, 3) nd (3, 1). Since the vlues of x could e in one of these intervls or the other, we once gin use the union symol to get {x x 6= 3} =( 1, 3) [ (3, 1). 3 ( 1, 3) [ (3, 1) (c) For the set {x x 6= ±3}, we proceed s efore nd exclude oth x = 3 nd x = 3 from our set. (Do you rememer wht we sid ck on 6 out x = ±3?) This reks the numer line into three intervls, ( 1, 3), ( 3, 3) nd (3, 1). Since the set descries rel numers which come from the first, second or third intervl, we hve {x x 6= ±3} = ( 1, 3) [ ( 3, 3) [ (3, 1). 3 3 ( 1, 3) [ ( 3, 3) [ (3, 1) (d) Grphing the set {x 1 < x pple 3 or x =5} yields the intervl ( 1, 3] long with the single numer 5. While we could express this single point s [5, 5], it is customry to write single point s singleton set, so in our cse we hve the set {5}. Thus our finl nswer is {x 1 < x pple 3 or x =5} =( 1, 3] [{5}. 1 3 5 ( 1, 3] [{5}
12 Prerequisites 2. We strt y grphing A =[ 5, 3) nd B = (1, 1) on the numer line. To find A\B, we need to find the numers in common to oth A nd B, in other words, the overlp of the two intervls. Clerly, everything etween 1 nd 3 is in oth A nd B. However, since 1 is in A ut not in B, 1 is not in the intersection. Similrly, since 3 is in B ut not in A, it isn t in the intersection either. Hence, A \ B = (1, 3). To find A [ B, we need to find the numers in t lest one of A or B. Grphiclly, we shde A nd B long with it. Notice here tht even though 1 isn t in B, it is in A, so it s the union long with ll the other elements of A etween 5 nd 1. A similr rgument goes for the inclusion of 3 in the union. The result of shding oth A nd B together gives us A [ B =[ 5, 1). 5 1 3 A =[ 5, 3), B = (1, 1) 5 1 3 A \ B = (1, 3) 5 1 3 A [ B =[ 5, 1)
0.1 Bsic Set Theory nd Intervl Nottion 13 0.1.3 Exercises 1. Find verl description for O = {2n 1 n 2 N} 2. Find roster description for X = {z 2 z 2 Z} 3 3. Let A = 3, 1.02, 5, 0.57, 1.23, p 3, 5.2020020002..., 20 10, 117 () List the elements of A which re nturl numers. () List the elements of A which re irrtionl numers. (c) Find A \ Z (d) Find A \ Q 4. Fill in the chrt elow. Set of Rel Numers Intervl Nottion Region on the Rel Numer Line {x 1 pple x < 5} [0, 3) {x 5 < x pple 0} 2 7 ( 3, 3) {x x pple 3} 5 7 ( 1, 9) {x x 3} 4
14 Prerequisites In Exercises 5-10, find the indicted intersection or union nd simplify if possile. Express your nswers in intervl nottion. 5. ( 1, 5] \ [0, 8) 6. ( 1, 1) [ [0, 6] 7. ( 1, 4] \ (0, 1) 8. ( 1, 0) \ [1, 5] 9. ( 1, 0) [ [1, 5] 10. ( 1, 5] \ [5, 8) In Exercises 11-22, write the set using intervl nottion. 11. {x x 6= 5} 12. {x x 6= 1} 13. {x x 6= 3, 4} 14. {x x 6= 0, 2} 15. {x x 6= 2, 2} 16. {x x 6= 0, ±4} 17. {x x pple 1 or x 1} 18. {x x < 3 or x 2} 19. {x x pple 3 or x > 0} 20. {x x pple 5 or x =6} 21. {x x > 2 or x = ±1} 22. {x 3 < x < 3 or x =4} For Exercises 23-28, use the lnk Venn Digrm elow A, B, nd C s guide for you to shde the following sets. U A B C 23. A [ C 24. B \ C 25. (A [ B) [ C 26. (A \ B) \ C 27. A \ (B [ C) 28. (A \ B) [ (A \ C) 29. Explin how your nswers to prolems 27 nd 28 show A\(B[C) =(A\B)[(A\C). Phrsed differently, this shows intersection distriutes over union. Discuss with your clssmtes if union distriutes over intersection. Use Venn Digrm to support your nswer.