1. Let A -(a, b) a, b Q,a b. Prove that A is denumerable. (You may cite any results from the text.) 2. Let SeRnE N) and define f:N-+S by n)- n + *. Since, by definition, S-f(N), it follows that f is onto (a) Show that f is one-to-one (b) Is S denumerable? Explain 3. Either prove or disprove each of the following. (You may cite any results from the text or other results from this assignment.) (a) If...
6) If E is any countable subset of real numbers prove that A*(E) = A*(E) = 0. 7) Show that the set of all real numbers IR is measurable with >(IR) = . 8) Prove that If f : [a, b] IR is continuous [a; b]then it is measurable [a, b]. 9) Give an example of a function f : [O, 1] IR which is measurable on [O, 1] but not continuos on [O, 1]. 10) Find the Lebesgue integral...
1. Prove that for any set S S R, S is closed if and only if Se is open. Notice the book has a proof of this, but it uses a different notation for set complements and a different definition of neighborhood. You may consult it, but you must write your proof using the definition for interior point I presented in class (also in the notes on blackboard). If you copy the proof from the book you will not receive...
Discrete Mathematics 22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1. What is the transitive closure of r? 23. Write an algorithm in pseudo code that converts numbers in decimal representation to octal (base 8) representation 24. Prove that the set of integers in countable 22. Let r be a relation on the integers such that (a, b) E r if and only if a +b 1....
4.1. Let S, TCR. Prove that (i). SCT S'CT. 4.2. Let S, TC R. Prove that (i. SCT SCT. 4.3. Show that if the set S is bounded above or below, then so is S' with the bounds of S and has greatest or smallest member accordingly, provided S' 0.
Let X be a metric space and let E C X. The boundary aE of E is defined by E EnE (a) Prove that DE = E\ E°. Here Eo is the set of all interior points of E; E° is called the interior of E (b) Prove that E is open if and only if EnaE Ø. (c) Prove that E is closed if and only if aE C E (d) For X R find Q (e) For X...
1. (Integers: primes, divisibility, parity.) (a) Let n be a positive integer. Prove that two numbers na +3n+6 and n2 + 2n +7 cannot be prime at the same time. (b) Find 15261527863698656776712345678%5 without using a calculator. (c) Let a be an integer number. Suppose a%2 = 1. Find all possible values of (4a +1)%6. 2. (Integers: %, =) (a) Suppose a, b, n are integer numbers and n > 0. Prove that (a+b)%n = (a%n +B%n)%n. (b) Let a,...
just trying to get the solutions to study, please answer if you are certain not expecting every question to be answered P1 Let PC 10, +00) be a set with the following property: For any k e Zso, there exists I E P such that kn s 1. Prove that inf P = 0. P2 Two real sequences {0,) and {0} are called adjacent if {a} is increasing. b) is decreasing, and limba - b) = 0. (a) Prove that,...
Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection between A and some infinite subset of N.) Problem 1. Let A be an infinite set such that |Al S INI. Prove A IN (Hint: First prove this for all infinite subsets B CN. Prove the general case by observing there is a bijection...
solve #5 only please 5 Prove that the function f in problem 4 is integrable and sf = 0. Suggestion: Use the suggestion for problem 4(a) to show that given €>0, there is a partition Pof [0, 1] with Uff, P) < 2€ , while Laf, P) =0. Do this by enclosing the points of the finite set where f(x) 2e in a finite set of disjoint closed intervals, each contained in (0,1), with the sum of the lengths <€....