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pivoxa15
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Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?
Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?
First, the "natural numbers" are NOT Z. The natural numbers include only the positive integers (some texts include 0) while Z is all integers. Obviously the natural numbers does NOT form a group since they do not have additive inverses. Typically, Z is defined in terms of natural numbers (say, as equivalence classes of pairs of natural numbers) and then the fact that they form an abelian group is proved.Do people usually prove that Z is an abelian group under (normal) addition or is it the definition of the natural numbers Z?
That + is abelian follows from
(x + y)z = xz + yz
x(y + z) = xy + xz (both proved by Peano i think)
Given a, b in Z.
(a + b)(1 + 1) = a(1 + 1) + b(1 + 1) = a + a + b + b,
(a + b)(1 + 1) = (a + b)1 + (a + b)1 = a + b + a + b
this implicates
a + b + a + b = a + a + b + b
so
a + b = b + a
so + is abelian
An axiom in what system? Certainly the distributive law is part of the definition of "ring" and so a axiom in that sense.