2.4 Sequences, Sequences of Sets

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1 72 CHAPTER 2. IMPORTANT PROPERTIES OF R 2.4 Sequeces, Sequeces of Sets Sequeces Defiitio (sequece Let S R. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For each K, we let x = f (. x is called the th term of the sequece f. For coveiece, we usually deote a sequece by {x } = 0 rather tha f. Some texts also use (x = 0. The startig poit, 0 is usually 1 i which case, we simply write {x } to deote a sequece. I theory, the startig poit does ot have to be 1. However, it is uderstood that whatever the startig poit is, the elemets x should be defied for ay 0. For example, if the geeral term of a sequece is x = 2 4, the, we must have 0 5. We ca thik of a sequece as a list of umbers. I this case, a sequece will look like: {x } = {x 1, x 2, x 3,...}. A sequece ca be give differet ways. { 1 1. List the elemets. For example, 2, 2 3, 3 } 4,.... From the elemets listed, the patter should be clear. 2. Give a formula to geerate the terms. For example, x = ( 1 2!. If the startig poit is ot specified, we use the smallest value of which will work. 3. A sequece ca be give recursively. The startig value of the sequece is give. The, a formula to geerate the th term from oe or more previous terms. For example, we could defie a sequece by givig: x 1 = 2 x +1 = 1 2 (x + 6 Aother example is the Fiboacci sequece defied by: x 1 = 1, x 2 = 1 x = x 1 + x 2 for 3 Like a fuctio, a sequece ca be plotted. However, sice the domai is a subset of Z, the plot will cosist of dots istead of a cotiuous curve. Sice a sequece is defied as a fuctio. everythig we defied for fuctios (bouds, supremum, ifimum,... also applies to sequeces. We restate those defiitios for coveiece.

2 2.4. SEQUENCES, SEQUENCES OF SETS 73 Defiitio (Bouded Sequece As sequece (x is said to be bouded above if its rage is bouded above. It is bouded below if its rage is bouded below. It is bouded if its rage is bouded. If the domai of (x is { Z : k for some iteger k} the the above defiitio simply state that the set {x : k} must be bouded above, below or both. Defiitio (Oe-to-oe Sequeces A sequece (x is said to be oeto-oe if wheever m the x x m. Defiitio (Mootoe Sequeces Let (x be a sequece. 1. (x is said to be icreasig if x x +1 for every i the domai of the sequece. If we have x < x +1, we say the sequece is strictly icreasig. 2. (x is said to be decreasig if x x +1 for every i the domai of the sequece. 3. A sequece that is either icreasig or decreasig is said to be mootoe. If it is either strictly icreasig or strictly decreasig, we say it is strictly mootoe Sequeces of Sets ad Idexed Families of Sets Sequeces of sets are similar to sequeces of real umbers with the exceptio that the terms of the sequece are sets. The otatio used is also similar. We also geeralize the defiitio to idexed families of sets where the idex set is ot a subset of Z but ay set. More precisely, we have: Defiitio Let X R, X ad let K = { N : 0 for some 0 N}. A sequece of subsets of X is a fuctio f : K P (X, the power set of X. For each K, we let A = f ( be a subset of X that is a elemet of P (x. Example Cosider {N } where N = {1, 2, 3,..., }. {N } is a sequece of subsets of N. { Example For each N, defie = x R : 0 < x < 1 } ( = 0, 1. { } is a sequece of subsets of R. The idex we use for the subsets does ot have to be a iteger. It ca be a elemet of ay set. I this case, istead of callig it a sequece of sets, we call it a idexed family of sets. We give a precise defiitio. Defiitio Let A ad X be o-empty sets. A idexed family of subsets of X with idex set s a fuctio f : A P (X (the power set of X. Like for sequeces, if f : A P (X, the for each α A, we let E α = f (α. We use a otatio similar to sequeces that is we deote the idexed family {E α } α A.

3 74 CHAPTER 2. IMPORTANT PROPERTIES OF R Remark A sequece of sets is just a idexed family of sets with N as idex set. Example For each x (0, 1, defie E x = {r Q : 0 r < x}. The, {E x } x (0,1 is a idexed family of subsets of Q. The idex set is (0, 1. The remaider of this sectio deals with sequeces of sets, though the results ad defiitios give ca be exteded to idexed families of subsets. Defiitio (Uio ad Itersectio of a Sequece of Subsets Let {A } be a sequece of subsets of a set X. 1. We defie = A 1 A 2... A = {x X : x for some atural umber 1 i } Similarly, we defie the uio of the etire sequece by = {x X : x for some atural umber i} 2. Similarly, we defie = A 1 A 2... A = {x X : x for every atural umber 1 i } ad = {x X : x for every atural umber i} Example Cosider {N } where N = {1, 2, 3,..., }. The, N ad N i = {1}. Example For each N, defie = we prove that I i = N i = { x R : 0 < x < 1 }. First,

4 2.4. SEQUENCES, SEQUENCES OF SETS 75 If this were ot the case, that is if we had x I i this would mea that o matter what is, x < 1 or 1 > x which cotradicts the Archimedea priciple (theorem We ext show that This because I 1 for ay 1. I i = I 1 Results about fiite itersectio ad uio of sets remai true i this settig. I other words, we have the equivalet of theorems (distributive properties, (De Morga s laws, (direct image of a set ad (iverse image of a set. We list the theorem here but leave their proof as exercises. Theorem (Distributive Laws Let E ad E be subsets of a set X. The, 1. E E i = (E E i 2. E E i = (E E i Proof. See problems. Theorem (De Morga s Laws Let {E } be a sequece of subsets of X. The, c 1. E i = c 2. E i = Proof. See problems. E c i E c i Theorem Let f : X Y 1. If {E } is a sequece of subsets of X, the f E i = f (E i f E i f (E i

5 76 CHAPTER 2. IMPORTANT PROPERTIES OF R 2. If {G } is a sequece of subsets of Y, the Proof. See problems. f 1 G i = f 1 (G i f 1 G i = f 1 (G i Defiitio (Cotractig ad Expadig Sequeces of Sets Suppose that (A is a sequece of sets. 1. We say that (A is expadig if we have A A +1 for every i the domai of the sequece. 2. We say that (A is cotractig if we have A +1 A for every i the domai of the sequece. ( Example Cosider the sequece of sets (J where J = 1, 1. ( J is a cotractig sequece as J +1 = J. 1 +1, 1 +1 Example Cosider the sequece of sets (K where K = (,. K is a expadig sequece as K K +1 = ( ( + 1, Exercises [ ] 1 1. Let =, 1. Evaluate 2. Let = 3. Let = [ 1 + 1, 5 2 ]. Evaluate ( 1 1, Evaluate

6 2.4. SEQUENCES, SEQUENCES OF SETS Prove theorem Prove theorem Prove theorem I particular, for part 2 of the theorem, explai why we do ot have equality. Give a ecessary coditio to have equality ad justify your aswer. 7. Let {A } be a expadig sequece of subsets of R. Prove that {R \ A } is a cotractig sequece. 8. Let {A } be a sequece of sets. (a Prove that if we defie i= for each, the sequece {B } is a cotractig sequece of sets. (b Prove that if we defie for each, the sequece {B } is a expadig sequece of sets. (c Prove that if we defie i= for each, the sequece {B } is a cotractig sequece of sets. (d Prove that if we defie for each, the sequece {B } is a expadig sequece of sets.

7 78 CHAPTER 2. IMPORTANT PROPERTIES OF R Hits for the Exercises For all the problems, remember to first clearly state what you have to prove or do. [ ] 1 1. Let =, 1. Evaluate Hit: Evaluate for a few values of to see what is happeig. You should the be able to cojecture what the aswer is. You the have to prove the cojecture which will ivolve provig two sets are equal. You will have to use the Archimedea priciple. 2. Let = [ 1 + 1, 5 2 ]. Evaluate Hit: Evaluate for a few values of to see what is happeig. You should the be able to cojecture what the aswer is. You the have to prove the cojecture which will ivolve provig two sets are equal. You will have to use the Archimedea priciple. 3. Let = ( 1 1, Evaluate Hit: Evaluate for a few values of to see what is happeig. You should the be able to cojecture what the aswer is. You the have to prove the cojecture which will ivolve provig two sets are equal. You will have to use the Archimedea priciple. 4. Prove theorem Hit: othig hard, just use defiitios ad stadard techiques to show two sets are equal. 5. Prove theorem Hit: othig hard, just use defiitios ad stadard techiques to show two sets are equal. 6. Prove theorem I particular, for part 2 of the theorem, explai why we do ot have equality. Give a ecessary coditio to have equality

8 2.4. SEQUENCES, SEQUENCES OF SETS 79 ad justify your aswer. Hit: othig hard, just use defiitios ad stadard techiques to show two sets are equal. 7. Let {A } be a expadig sequece of subsets of R. Prove that {R \ A } is a cotractig sequece. Hit: o diffi culty, just use defiitios. Make sure to first clearly write what you have to prove. 8. Let {A } be a sequece of sets. (a Prove that if we defie i= for each, the sequece {B } is a cotractig sequece of sets. Hit: Make sure you uderstad how this set B is defied by makig your ow examples. You prove the result by just usig defiitios. Agai, first state clearly what you have to prove. (b Prove that if we defie i= for each, the sequece {B } is a expadig sequece of sets. Hit: Make sure you uderstad how this set B is defied by makig your ow examples. You prove the result by just usig defiitios. Agai, first state clearly what you have to prove. (c Prove that if we defie for each, the sequece {B } is a cotractig sequece of sets. Hit: Make sure you uderstad how this set B is defied by makig your ow examples. You prove the result by just usig defiitios. Agai, first state clearly what you have to prove. (d Prove that if we defie for each, the sequece {B } is a expadig sequece of sets. Hit: Make sure you uderstad how this set B is defied by makig your ow examples. You prove the result by just usig defiitios. Agai, first state clearly what you have to prove.