please send handwritten solution for b To submit Let S be the setS= {a+bk : a,b € R},where k is a formal symbol. Define addition and multiplication operations on S asfollows: given elements x = a+bk and y =c+dk in S.x+y:= (a+c)+(b+d)k,xy:= (ac +bd) +(ad +be)k.(a) Prove both identity laws for S. Include a short explanation (one sentence is fine)of how you know what the identity elements are(b) Prove that the multiplicative inverse law is false for S. [That is, don't just writedown a counterexample; also prove that your counterexample is valid.]

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Answered: Prove that the multiplicative inverse…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.4: Binary Operations

Problem 6E

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Prove that the multiplicative inverse law is false for S. [That is, don't just write down a counterexample; also prove that your counterexample is valid.]