Fast plz Show that given statement , no typing If S is a non-empty set of real numbers which is boundedabove, then a real number s is the supremum of S if and only if the followingtwo conditions hold :(i) xSs V xeS.(ii) Given.any ɛ> 0, 3 some x e S such that x > s - E.11

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Answered: If S is a non-empty set of real numbers…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.1: Sets

Problem 13E: 13. Let Z denote the set of all integers, and let Prove that .

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13. Let Z denote the set of all integers, and let Prove that .
25. Prove that if and are integers and, then either or. (Hint: If, then either or, and similarly for. Consider for the various causes.)
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.
16. Let and define on by if and only if . Determine whether is reflexive, symmetric, or transitive.
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 .
12. Let Z denote the set of all integers, and let Prove that .
In each of the following parts, a relation R is defined on the power set (A) of the nonempty set A. Determine in each case whether R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if xy. b. xRy if and only if xy.

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If S is a non-empty set of real numbers which is bounded above, then a real number s is the supremum of S if and only if the following two conditions hold : (i) x 0, 3 some x e S such that x > s- E.