[10] Let E : N→{0, 1}∗ be a total one-to-one function whose range is prefix-free. E defines a prefix-code. Define the mapping · :

N ×N →N by x, y = E(x)y.

(a) Show that · is total and one-to-one.

(b) Show that we can extend this scheme to k-tuples (n1, n2,...,nk) of natural numbers to obtain a total one-to-one mapping from N ×N ×

···×N into N .

Comments. Define the mapping for (x, y, z) as x,y, z , and iterate this construction.