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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
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter1: Fundamentals
Section1.4: Binary Operations
Problem 14E: Assume that is a binary operation on a non empty set A, and suppose that is both commutative and...
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(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.
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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