A set of real numbers D is called dense if and only if every open interval contains at least one point of D. Assume a set E and its complement E = {x € R : x ¢ E} are both dense. Let f : [0, 1] → R be given by f(x) = 1 if x E E and f(x) = 0 if x ¢ E. (Thus f(x) = XE (x) for x E [0, 1].) Prove that f is not integrable on [0, 1].

A set of real numbers D is called dense if and only if every open interval contains at least one point of D. Assume a set E and its complement E = {x € R : x ¢ E} are both dense. Let f : [0, 1] → R be given by f(x) = 1 if x E E and f(x) = 0 if x ¢ E. (Thus f(x) = XE (x) for x E [0, 1].) Prove that f is not integrable on [0, 1].

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.2: Mappings

Problem 23E: Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove...

See similar textbooks

Similar questions

Let f(x),g(x),h(x)F[x] where f(x) and g(x) are relatively prime. If h(x)f(x), prove that h(x) and g(x) are relatively prime.
31. Prove statement of Theorem : for all integers and .
Let a and b be constant integers with a0, and let the mapping f:ZZ be defined by f(x)=ax+b. Prove that f is one-to-one. Prove that f is onto if and only if a=1 or a=1.
Complete the proof of Theorem 5.30 by providing the following statements, where and are arbitrary elements of and ordered integral domain. If and, then. One and only one of the following statements is true: . Theorem 5.30 Properties of Suppose that is an ordered integral domain. The relation has the following properties, whereand are arbitrary elements of. If then. If and then. If and then. One and only one of the following statements is true: .
A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.
For an element x of an ordered integral domain D, the absolute value | x | is defined by | x |={ xifx0xif0x Prove that | x |=| x | for all xD. Prove that | x |x| x | for all xD. Prove that | xy |=| x || y | for all x,yD. Prove that | x+y || x |+| y | for all x,yD. Prove that | | x || y | || xy | for all x,yD.
2. Prove the following statements for arbitrary elements of an ordered integral domain . a. If and then . b. If and then . c. If then . d. If in then for every positive integer . e. If and then . f. If and then .
Let be as described in the proof of Theorem. Give a specific example of a positive element of .
Label each of the following statements as either true or false. The Well-Ordering Theorem implies that the set of even integers contains a least element.
26. Let and. Prove that for any subset of T of .