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malaygoel · #2 July 1st, 2009, 09:03 AM

Quote:

Originally Posted by alexandros

given a set with the symbols :
" + " for addition
" - " for inverse
the constants :
0
AND the axioms:
for all a,b,c : a+(b+c) = (a+b) + c
for all a : a+0 = a
for all a : a +(-a) =0
for all a,b : a+b = b+a
PROVE using CONTRADICTION
1) THE uniqness of zero
2) THE uniqness of the inverse

1) Let $0^,$ be a number other than zero such that for all a,

$a+0^,=a$

then, since zero is number

$0+0^,=0$

also by our fourth axiom, $0+0^,=0^,+0$ which is $0^,$ by our second axiom.

which implies $0^,$ is same as $0$ ....hence, zero is unique.