(b) 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.]
(b) 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.]
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.1: Definition Of A Group
Problem 43E: Write out the elements of P(A) for the set A={ a,b,c }, and construct an addition table for P(A)...
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please send handwritten solution for part b
![To submit Let S be the set
S= {a+bk : a,b € R},
where k is a formal symbol. Define addition and multiplication operations on S as
follows: 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 write
down a counterexample; also prove that your counterexample is valid.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0f8892e-30c8-410c-bb56-3f9903ad2c7b%2F84045cf9-94c1-4114-8773-7fa1d991c8d0%2Fhlfmgrh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:To submit Let S be the set
S= {a+bk : a,b € R},
where k is a formal symbol. Define addition and multiplication operations on S as
follows: 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 write
down a counterexample; also prove that your counterexample is valid.]
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