2.7 Sequences, Sequences of Sets


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1 2.7. SEQUENCES, SEQUENCES OF SETS Sequeces, Sequeces of Sets Sequeces Defiitio 190 (sequece Let S be some set. 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 68 CHAPTER 2. THE STRUCTURE OF R Defiitio 191 (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 192 (Oetooe Sequeces A sequece (x is said to be oetooe if wheever m the x x m. Defiitio 193 (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 Defiitio 194 Let A ad X be oempty sets. dexed 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. If A = N as it is ofte the case (it will be for us, the {E α } α N is deoted {E } =1 or simply {E } ad it is called a sequece of subsets of X. Example 195 Cosider {N } =1 where N = {1, 2, 3,..., }. {N } =1 is a sequece of subsets of N. Example 196 For each N, defie I = {I } =1 is a sequece of subsets of R. { x R : 0 < x < 1 } ( = 0, 1. Example 197 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 198 (Uio ad Itersectio of a Sequece of Subsets Let {A } be a sequece of subsets of a set X.
3 2.7. SEQUENCES, SEQUENCES OF SETS We defie = A 1 A 2... A = {x X : x for some iteger 1 i } Similarly, we defie the uio of the etire sequece by = {x X : x for some iteger i} 2. Similarly, we defie = A 1 A 2... A = {x X : x for every iteger 1 i } ad = {x X : x for every iteger i} Example 199 Cosider {N } =1 where N = {1, 2, 3,..., }. The, ad N i = {1}. Example 200 For each N, defie I = prove that I i = If this were ot the case, that is if we had x N i = N { x R : 0 < x < 1 }. First, we I i this would mea that o matter what is, x < 1 or 1 > x which cotradicts the Archimedea priciple (theorem 173. We ext show that This because I I 1 for ay 1. I i = I 1
4 70 CHAPTER 2. THE STRUCTURE OF R Results about fiite itersectio ad uio of sets remai true i this settig. I other words, we have the equivalet of theorems 16 (distributive properties, 20 (De Morga s laws, 60 (direct image of a set ad 62 (iverse image of a set. We list the theorem here but leave their proof as exercises. Theorem 201 (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 202 (De Morga s Laws Let {E } be a sequece of subsets of X. The, ( c E i = ( c E i = E c i E c i Proof. See problems. Theorem 203 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 2. If {G } is a sequece of subsets of Y, the f 1 G i = f 1 (G i f 1 G i = f 1 (G i
5 2.7. SEQUENCES, SEQUENCES OF SETS 71 Proof. See problems. Defiitio 204 (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 Exercises [ ] 1 1. Let I =, 1. Evaluate 2. Let I = 3. Let I = 4. Prove theorem Prove theorem 202. =1 [ 1 + 1, 5 2 ]. Evaluate =1 ( 1 1, Evaluate =1 6. Prove theorem 203. 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. Explai why if {A } is a expadig sequece of subsets of R, the {R \ A } is a cotractig sequece. 8. Let {A } be a sequece of sets. (a Prove that if we defie I I I B = for each, the sequece {B } is a cotractig sequece of sets. i=
6 72 CHAPTER 2. THE STRUCTURE OF R (b Prove that if we defie B = for each, the sequece {B } is a expadig sequece of sets. i= (c Prove that if we defie B = for each, the sequece {B } is a cotractig sequece of sets. (d Prove that if we defie B = for each, the sequece {B } is a expadig sequece of sets.
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