The statement 0.x = 0 is not an axiom and can thus be proved.
The axioms for the integers are, for addition
(A1) a + (b + c) = (a + b) + c
(A2) a + 0 = a
(A3) a + (-a) = 0
(A4) a + b = b + a
for multiplication
(M1) a(bc) = (ab)c
(M2) 1a = a
(M3) ab = ba
and for addition and multiplication ('distributive laws')
(D1) a(b+c)=ab+ac
(D2) (a+b)c=ac+ab.
So how do mathematicians prove that 0.a=0?
They do something like this.
a = a
a = (1+0).a (by A2 and M2)
a = 1.a + 0.a (by D2)
a = a + 0.a (by M2)
0.a = 0 by (A2).
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I've forgotten what the correct word is for this type of nit-picking.
ReplyDeleteYou should have had A3 in the last statement:)
Ray
Or: (0 + 0)a = 0a; so 0a + 0a = 0a, hence 0a = 0.
ReplyDelete