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
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
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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please send handwritten solution for part a
![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%2F4e5527bb-ef8e-4371-930b-b8f502050d12%2F3dxh4kn_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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