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Define $\oplus$ on $\mathbb{R} \times \mathbb{R}$ by setting $$(a,b) \oplus (c,d)=(ac-bd, ad+bc).$$ How to show that $(\mathbb{R} \times \mathbb{R}, \oplus)$ is an algebraic system. I don't understand the difference between algebraic structure andalgebraic system I read this article but I didnt understand it properly. Can anybody help me me understand by giving hints?

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    It's generally considered rude here to ask questions in the imperative. You might get better responses if you phrase them in the form "I have a question about this problem: ..." or something similar.2012-08-16
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    What do you mean by "algebraic system"? Do you have a specific definition in mind? What have you tried so far?2012-08-16
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    continued from [here](http://openstudy.com/study#/updates/502d3ce5e4b0e5faacce8e07)2012-08-16
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    @CalvinMcPhail-Snyder I am new to this site and not sure what is required and what are the requirements. Well the only algebraic structure that I am familiar with is a group. So I am assuming we can show that it is a group2012-08-16
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    @math101 No problem! Just letting you know for the future.2012-08-16
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    What have you done so far? What are you stuck on? You specified conditions it should have to satisfy your definition. Have you worked on demonstrating that any of those conditions are satisfied?2012-08-16
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    @EdGorcenski as the answerer has shown it isnt a group. I havent covered rings or fields SO i am unsure what are the requirements of an algebraic system. Therefore I am a little bit confuseddd2012-08-16

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$\oplus$ on $\mathbb{R} \times \mathbb{R}$ is a just multiplication of the complex numbers on $\mathbb{C}$. It is not a group since $(0,0) \in \mathbb{R} \times \mathbb{R}$ does not have a inverse under $\oplus$.

However $\mathbb{R} \times \mathbb{R} - \{0\}$ is a group under $\oplus$.


$(\mathbb{R} \times \mathbb{R}, \oplus)$ however is a commutative monoid. It is associative, has an identity element, but not every element has an inverse.

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    Are [algebraic structure](http://en.wikipedia.org/wiki/Algebraic_structure) and [algebraic system](http://planetmath.org/AlgebraicSystem.html) same?2012-08-16
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    @experimentX Both term algebraic system and algebraic structure are not generally used. In logic and model theory, I have heard the term algebraic language as a language with no relation symbols, and a algebraic structure as a structure in an algebraic language. Groups, vector spaces, rings are usually given an algebraic language. They are hence algebraic structures.2012-08-16
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    Oh, this is just simple undergraduate algebra. anyway thanks for response +12012-08-16