Prove that the recursive algorithm for finding the concatenation of i copies of a bit string that you gave in Exercise 38 is correct.
MODULE 14: INDUCTION AND RECURSIVE DEFINITION/ALGORITHM Induction/Recursive Definition Algorithm Chapter Summary Mathematical Induction Strong Induction Well-Ordering Recursive Definitions Structural Induction Recursive Algorithms → 5.1 Mathematical Induction ← Climbing an Infinite Ladder Suppose we have an infinite ladder: 1. We can reach the first rung of the ladder. 2. If we can reach a particular rung of the ladder, then we can reach the next rung. From (1), we can reach the first rung. Then by applying (2), we can reach the second rung. Applying (2) again, the third rung. And so on. We can apply (2) as many times as we feel to reach any particular rung no matter the height. This is proof by mathematical induction. Principle of Mathematical Induction “To prove that P(n) is true for all positive integers n, we complete these steps: Basis Step: Show that P(1) is true. Inductive Step: Show that P(k) → P(k + 1) is true for all positive integer k. To complete the inductive step, assuming the INDUCTIVE HYPOTHESIS that P(k) holds for an arbitrary integer k, show that P(k + 1) must be true.” Looking at the “Climbing an Infinite Ladder” example: BASIS STEP: By (1), WE CAN reach rung 1. INDUCTIVE STEP: Assume the inductive hypothesis that we can reach rung k. Apply (2) and we can reach rung k + 1. Hence, P(k) → P(k + 1) is true for all positive integers k. We can reach every rung on the ladder. Important Points about MI (Math