(03/06/2011, 08:52 AM)bo198214 Wrote: PS: at a[n]1 = a you should add n>1Indeed, thanks for checking

1. a[2]b = a * b

Proof:

For b = 1:

By definition a[2]1 = a = a * 1

If a[2]b = a * b for a given b, then we wish to prove that a[2](b + 1) = a * (b + 1):

a[2](b + 1) = a[1](a[2]b) = a + (a[2]b) = a + (a * b) = a * (b + 1)

So it is proven by induction.

2. a[3]b = a ^ b

Proof:

For b = 1:

By definition a[3]1 = a = a ^ 1

If a[3]b = a ^ b for a given b, then we wish to prove that a[3](b + 1) = a ^ (b + 1):

a[3](b + 1) = a[2](a[3]b) = a * (a[3]b) = a * (a ^ b) = a ^ (b + 1)

So it is proven by induction.