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When I followed an introductory! course group theory and throughout all my Math courses as a physicist, subtraction was always defined in terms of the inverse element and addition.

Is this the only way? I.e.: can subtraction be defined without addition?

If this is too broad a question, perhaps focus on groups and other physically relevant concepts.

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    Here's teh start of a bunch of C++ programmers discussing the question: http://chat.stackoverflow.com/transcript/message/4935773#4935773, where I held that I believe that the axioms of math are simpler if subtraction is defined in terms of addition, but I'm not a math guy, so we brought it here.2012-08-14
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    Addition is more fundamental in the sense that the natural numbers (the most fundamental set of numbers) is closed under addition but not subtraction.2012-08-14
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    And *appending* the negative numbers and zero to the natural numbers obviates subtraction: $a-b = a+(-b)$2012-08-14
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    Fundamental to what?2012-08-14
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    We start with two statements "0", "a+b" and "a-b" which we *understand* and want to translate into one another. a+b=a-(0-b), a-b=a+(-b) addition is defined using only subtraction and subtraction isnt defined using anly addition, because to define -b, in terms of "a+b" and "a-b" statements we have to write -b=0-b, so writing a-b=a+(-b) isnt very fundamental ti should be a+(0-b). Both equation needs definition of 0.So subtraction is more fundamental2012-08-14
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    @Joachim writes that subtraction is not even an operation, that is because subraction is something more general, now we could argue if more general means more fundamental :D And that is fundamental reason way my above translation is possible ;) (But there are some hiden assumptions in my post;)2012-08-14
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    Whoa! This is a group theory thread! Why has it yet to be pointed out that $a\mapsto -a$ (for all $a\in A$) is an isomorphism?!?2012-08-15
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    @user1729 in my defense, I touched that aspect already [in this comment](http://math.stackexchange.com/questions/182526/is-addition-more-fundamental-than-subtraction#comment420762_182528)2012-08-15
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    @Qbik: that's backwards, you've shown that if you can subtract you can add, so addition is sort of more often possible (since sometimes you can add but you can't subtract) - i.e. more general.2012-08-15
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    @Ben Millwood good point !!! :) ps. I cant edit my comments, and there are some misstakes2012-08-15

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