Theorem 6.8: (The Associative Property for Multiplication of Natural Numbers): Let a, b, and c be natural numbers. Then

a (bc ) = (ab )c

Proof:

a (bc ) = #(a x (b x c )),

and

(ab )c = #((a x b ) x c ).

We will establish a one to one correspondence between a x ( b x c ) and (a x b ) x c. Define

by

f((x , (y , z ))) = ((x , y), z).

To show that f is one to one, suppose that

f(x , (y , z)) = f(u , (v , w))

Then by the definition of f,

((x , y ), z ) = ((u , v ), w )

so by Theorem 1.7

(x , y) = (u , v) and z = w.

Since

(x , y) = (u , v)

x = u and y = v.

also by Theorem 1.7. Thus

(x , (y , z)) = (u , (v , w))

To show that it is onto, let

Then

((x , y ), z ) = f ((x , (y , z )))

Since f is a one to one correspondence,

#(a x (b x c )) = #((a x b ) x c )

by Theorem 5.5.