Theorem 6.7: (The Commutative Property for Multiplication of Natural Numbers): Let a and b be two natural numbers. Then

ab = ba

Proof: ab = #(a x b ). ba = #(b x a ), so by Theorem 5.5, it will suffice to establish a one to one correspondence between a x b and b x a. Define

by

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

To show that f is one to one, suppose that

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

then, by the definition of f,

(y , x ) = (v , u )

so

y = v and x = u

by Theorem 1.7, and then

(x , y ) = (u , v )

To show that it is onto, let

Then

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