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May 13th, 2011, 11:44 AM  #1 
Newbie Joined: May 2011 Posts: 2 Thanks: 0  Is an algebraic structure a set?
A set is defined as a collection of 'objects'. Does there exist something that is not considered an object in mathmatics? The reason why I ask is because when I look up the definition of a ring, it says a ring is an 'algebraic structure' existing of a set and 2 binary operations. Why can't one say a ring is a set that includes 2 binary operations? Do they give this definition because depicting an object of a set neglects semantics? If that was true then we wouldn't be able to calculate anything with numbers, because numbers are just an agreed representation of a concept communicated through informal human language, right? 
May 14th, 2011, 02:20 AM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Is an algebraic structure a set?
You have to decide what the binary operations do, so you have to define x+y and x*y for all x and y in the set. That's where the algebraic structure comes in. The set doesn't itself include the binary operations. But you can represent an algebraic structure as a set: for example, {(1,+,1,2), (1,+,2,3), ...} is a set of ordered quadruples. 

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algebraic, set, structure 
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