This explains why we usually consider congruence modulo m >1 only.Problem 9. In each case determine if the given relation R on A is an equivalence relation.Justify your answer.(a) A = Z and a Rb if ab > 0.(b) A = Z and aRb if 5|a – b.(c) A = Z and aRb if 5 a2 – b².(d) A = Z and a Rb if 5|a3 – 63.(e) A = Z and aRb if 5 a4 - b4.(f) A = Z and aRb if 5 a -65.(g) A = R? and (a, b)R(c, d) if a = c.(h) A = R? and (a, b)R(c, d) if a² + b² = c² + d.Problem 10. In each case determine if the given relation R on A is a partial order. JustifyIf the answer is positive determine if the order is total: if a b E A then aBh orvour answer

Note:- As per our guidelines, we can answer the first three sub-parts of this problem. Please…

Problem 9. In each case determine if the given relation R on A is an equivalence relation. Justify your answer. (a) A = Z and aRb if ab > 0. (b) A = Z and aRb if 5|a – b. (c) A = Z and aRb if 5 a? - 6². -

Problem 9. In each case determine if the given relation R on A is an equivalence relation. Justify your answer. (a) A = Z and aRb if ab > 0. (b) A = Z and aRb if 5|a – b. (c) A = Z and aRb if 5 a? - 6². -

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 4E: 4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if...

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Question

This explains why we usually consider congruence modulo m >1 only.
Problem 9. In each case determine if the given relation R on A is an equivalence relation.
Justify your answer.
(a) A = Z and a Rb if ab > 0.
(b) A = Z and aRb if 5|a – b.
(c) A = Z and aRb if 5 a2 – b².
(d) A = Z and a Rb if 5|a3 – 63.
(e) A = Z and aRb if 5 a4 - b4.
(f) A = Z and aRb if 5 a -65.
(g) A = R? and (a, b)R(c, d) if a = c.
(h) A = R? and (a, b)R(c, d) if a² + b² = c² + d.
Problem 10. In each case determine if the given relation R on A is a partial order. Justify
If the answer is positive determine if the order is total: if a b E A then aBh or
vour answer

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