Consider the following recursively defined set S Basis elements: {2, 4, 8} • Recursive step 1: x, y e S → x * y e S Recursive step 2: re S → 2 • x € S Use structural induction to show that all elements in S are powers of two. Formally, x is a power of two if and only if: 3keN(x = 2k)

Prove that multiplication on N preserves order: (a) If a = b,then ac = bc. (b) If a < b,then ac < bc.

ANSWER ANY FOUR (4) QUESTIONS FROM THIS SECTION 1. a) Suppose that fis defined recursively by: f (0) = 5 and f(n+1) = 2 fn+ 5. Find f(1), (2), (3) and (4)? b) For which positive integer n is it true that 2">n3? c) Prove your answer in (b) above using mathematical induction. d) Give a recursive definition of the sequence {an}, n= 1, 2, 3... if an= 2n + 1 e) Use your definition in (d) above to find a10, and ais f) Let A = {1, 2, {{1,2}}}. Find the power set P(A) File DISCRETE STRUCTURES FINAL EXAMS.pdf Page 2 of 5