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?
How to show that this is an Algebraic System
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$\begingroup$
abstract-algebra
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0@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 confuseddd – 2012-08-16
1 Answers
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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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0Oh, this is just simple undergraduate algebra. anyway thanks for response +1 – 2012-08-16