please help me on question 14 ve and symmetric but not tran-{yE A: x R y}. (Note that R(x) is the same as x except(A) =that R is not an equivalence relation in this exercise.) Does the set A ={R(x): x E A } always form a partition of A? Prove that your answer is correct.12.Repeat Exercise 11, assuming R is reflexive and transitive but not symmetric.no nou.13.Repeat Exercise 11, assuming R is symmetric and transitive but not reflexive.vib ai 14. Let A be a set with at least three elements.(a) If P =of A? Explain. What if B, = B,?g ow doidw(b) If P = {B1, B2, B3} is a partition of A, is {Bº, B§, B§} a partition of A?Explain. Consider the possibility that two or more of the elements of Pmay be equal.(c) If P = {B1, B2} is a partition of A, 6, is a partition of B1, C2 is a parti-{B1, B2} is a partition of A with B, + B,, is {B“, B5} a partitionGG C%3D29%3DCombneFradeDo 15. Assign a grade of A (correct), C (partially correct), or F (failure) to each.slugmoo w dsnWgtion of B,, and B # B,, prove that C, U6, is a partition of A.1yodi Jud oot Justify assignments of grades other than A.(GLGUCG(a) Claim. Let R be an equivalence relation on the set A, and let x, y, andz be elements of A. If x E y and z ¢ x, then z ¢ ỹ.un diw oilomdiis ol,Ini noqu bnoqob Jeri «Proof.9101 210w oolo nuitivity, v Rz. so z E v. Therefore, if xe y and z x, then z & y.E 1odi yea biuAssume that xe ỹ and z e x. Then y R x and x R z. By tran-Let R be an equivalence relation on the set A, and let x, y, and(b) Claim.(o bom) zbe elements of A. If x e ỹ and z ¢ x, then z ¢ ỹ."Proof." Assume that xE y, and assume that z E y. Then y R x and99y R z. By symmetry, x R y, and by transitivity, x R z. Therefore, z e I.and z & X. then z ¢ y.

ng R is symmetric and transitive but not reflexive. 14. Let A be a set with at least three elements. abivib lo so (a) If P = {B1, B2} is a partition of A with B, + B,, is {B{, B5} a partition 1 ei of A? Explain. What if B, = B,?gon ovdoie (b) If P = {B1, B2, B3} is a partition of A, is {B{, B5, B§} a partition of A? Explain. Consider the possibility that two or more of the elements of P may be equal.omdi (c) If P = {B1, B2} is a partition of A, C, is a partition of B1, C2 is a parti- coubnanon oolo bolleo dgim luqmow denw g tion of B, and B¡ + B,, prove that C, U C, is a partition of A. bort

ng R is symmetric and transitive but not reflexive. 14. Let A be a set with at least three elements. abivib lo so (a) If P = {B1, B2} is a partition of A with B, + B,, is {B{, B5} a partition 1 ei of A? Explain. What if B, = B,?gon ovdoie (b) If P = {B1, B2, B3} is a partition of A, is {B{, B5, B§} a partition of A? Explain. Consider the possibility that two or more of the elements of P may be equal.omdi (c) If P = {B1, B2} is a partition of A, C, is a partition of B1, C2 is a parti- coubnanon oolo bolleo dgim luqmow denw g tion of B, and B¡ + B,, prove that C, U C, is a partition of A. bort

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 22E: A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which...

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