Let S be the set S= {a+bk: a,bƐ 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 + bc)k.
Let S be the set S= {a+bk: a,bƐ 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 + bc)k.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.4: Binary Operations
Problem 6E
Related questions
Question
(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.]
![Let S be the set
S= {a+bk: a,bƐ 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 + bc)k.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F283b3a0d-0821-472c-a1ce-fcf162c3f2c6%2F3a423176-7202-407a-be60-b5adf1c3fa18%2Fdhzae38_processed.png&w=3840&q=75)
Transcribed Image Text:Let S be the set
S= {a+bk: a,bƐ 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 + bc)k.
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