Let S be the set of all Cauchy sequences (sn) such that sn є Q for...
problem 23 please :) and here is Q.21 Problem 23. Recall from Problem 21 the equivalence relation ~ on the set of rational Cauchy sequences C. Define 〈z) E C to be eventually positive if there is an M є N such that xn > 0 for all Prove that eventually positive is a well defined notion on c/ (z〉 ~ 〈y), then 〈y〉 İs eventually positive. ie. if 〈z) is eventually positive and Problem 21. Let C be the...
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
Let Mi be the set of all sequences {a.);, of real num bers such that Σ converges. More formally, we could write this as 1 lal M1a :(W) ai R and i=1 We introduce a function p: Mi x MiR by setting 95 Let (Mi,p) denote the particular metric space we introduced above, and for each X = {xīた1 e M and for each i, we refer to the number xi as the ith coordinate of X. For each N...
#9. all one problem. then e is 0U tric space. Show that there is an isometric imbedding h of X 、D), as follows: Let X denote the set of all space () into a complete metric Cauchy sequences of points of X. Define x~ y if Let [x] denote the equivalence class of x; and let Y denote the set of these eq x (xi, x2, ...) uiv alence classes. Define a metric D on Y by the equation linnod(xn,...
Please answer all parts. Thank you! 20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class...
Please answer all parts. (2) (a) Give an example of sequences (sn) and (tn) such that lim sn ntoo 0, but the sequence (sntn) does not converge does not converge.) (b) Let (sn) and (tn) be sequences such that lim sn (Prove that it O and (tn пH00 is a bounded sequence. Show that (sntn) must converge to 0. 1 increasing subsequence of it (b) Find a decreasing subsequence of it (3) Consider the sequence an COS (а) Find an...
only the last question plz 2. Let S be a set and let ~ be an equivalence relation on it. Let π denote the canonical projection, T: S → S/ ~, π(x) = [x] . Prove that π is an onto map. Give an example of a set and relation for which π is not one-to-one. What is the necessary and sufficient condition on ~ for π is one-to-one? (State your answers and prove them 2. Let S be a...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
Define where S is the collection of all real valued sequences i.e. S = {x : N → R} and we denote xi for the ith element a the sequence x E S. Take for any x EL (i) Show that lic 12 (where recall 1-(x є s i Izel < oo)) (ii) Is l? Prove this or find a counterexample to show that these two sets do not coinside (iii) ls e c loc where recall looー(x є sl...
Correction: first problem is #2, not #1. Please show all steps in the proofs. Definitions for problems #2 through #5: Let C be the set of all Cauchy sequences of rational numbers, with the operations of addition and multiplication defined on C by (an) + (bn) = (an + bn) and (an)(bn) = (anbn). Let N be the subset of C consisting of all null sequences in c. Properties of a ring: A1. (a + b) +c= a + b...