Let ≥ be a relation defined on sets A and B by A ≥ B iff there exists a surjective functionf : A → B. Show that ≥ is not an equivalence relation. (a) Let > be a relation defined on sets A and B by A > B iff there exists a surjective functionf: A → B. Show that > is not an equivalence relation. (Hint: Try to find a counter exampleto one of the properties of an equivalence relation)(b) Prove the following:n(0.) =rE(1,00)(c) Is the set (0, 1) U {1,2, 3, .., 100} countable? Justify your answer.(d) Let A CR. If A is bounded above and contains one of its upper bounds, prove that this upperbound is the supremum of A. (Hint: Try a proof by contraction)

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Answered: Let ≥ be a relation defined on sets A…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 28E

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Label each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.
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 .
In each of the following parts, a relation is defined on the set of all human beings. Determine whether the relation is reflective, symmetric, or transitive. Justify your answers. xRy if and only if x lives within 400 miles of y. xRy if and only if x is the father of y. xRy if and only if x is a first cousin of y. xRy if and only if x and y were born in the same year. xRy if and only if x and y have the same mother. xRy if and only if x and y have the same hair colour.
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.
Give an example of a relation R on a nonempty set A that is symmetric and transitive, but not reflexive.
In Exercises , prove the statements concerning the relation on the set of all integers. 18. If and , then .
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.
True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.

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Let ≥ be a relation defined on sets A and B by A ≥ B iff there exists a surjective function f : A → B. Show that ≥ is not an equivalence relation.