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May 13th, 2011, 11:44 AM   #1
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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?
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May 14th, 2011, 02:20 AM   #2
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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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