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 +bc)k. 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,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 +bc)k. 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.]

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

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 +bc)k.

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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