answer it asap and correctly thank you A and B are subsets of R (set of reals)Let A be a non empty subset of a boundedset B. why does inf A and sup A exist?Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B.

Answered: A and B are subsets of R (set of reals)…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.1: Sets

Problem 12E: 12. Let Z denote the set of all integers, and let Prove that .

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Similar questions

Let A be a set of integers closed under subtraction. a. Prove that if A is nonempty, then 0 is in A. b. Prove that if x is in A then x is in A.
12. Let Z denote the set of all integers, and let Prove that .
16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
Prove that if and are real numbers such that , then there exist a rational number such that . (Hint: Use Exercise 25 to obtain such that . Then choose to be the least integer such that . With these choices of and , show that and then that .) If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)
If and are positive real numbers, prove that there exist a positive integer such that . This property is called Archimedean Property of the real numbers. (Hint: If for all , then is an upper bound for the set . Use the completeness property of to arrive at a contradiction.)
21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.
13. Consider the set of all nonempty subsets of . Determine whether the given relation on is reflexive, symmetric or transitive. Justify your answers. a. if and only if is subset of . b. if and only if is a proper subset of . c. if and only if and have the same number of elements.
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.

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A and B are subsets of R (set of reals) Let A be a non empty subset of a bounded set B. why does inf A and sup A exist? Show that (a) inf B ≤ inf A and (b) sup A ≤ sup B.