Let S be the set of ordered pairs of integers defined by the

Let S be the set of ordered pairs of integers defined by the following recursive definition. Base Step: (2, 5) element of S. Recursive Step: If (x1, x2) element of S and (y1, y2) element of S, then (x1y1, x2 + y2) element of S. Prove that S = {(2^k, 5k)|k element of Z^+}.

Solution

(2,5) belong to S

Hence (x1y1, x2+y2) belong to S

Substitute x 1 = y1=2 and x2 = y2 = 5

Then (2(2), 5+5) = (4,10) belong to S

Again consider (2,5) and (4,10)

Then (8, 15) belong to S

Repeat this process k times.

We get (2^k,5k) belong to S

Hence proved