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
2) The inverse of is . Assume that there exists another inverse element denoted as such that
Now, by the definition of inverse element, we have:
But from the third axiom, we have:
Ergo:
Which contradicts our original assumption , so the inverse element must be unique.