(Pn): For all r with 1 ≤ r < n, there is no one-to-one function from Nn to Nr.

Let Nk= {1,2,…,k} be a finite set with k elements. This problems shows that if n, r elements of N with n > r, then there is no one-to-one function from Nn to Nr. We prove this by induction on n = 2 for the following statement.

(Pn): For all r with 1 = r < n, there is no one-to-one function from Nn to Nr.

a. The first step covers the general case when r=1 and any n>r.

Prove that if n > r=1, then there is no one-to-one function from Nn to Nr.