2.6 Applications Of Induction & other ideas Important Theorems Midwestern State University – Computer Science Dr. Ranette Halverson

Theorem 2.8 – Binary Search For any non-negative integer n, at most n+1 comparisons are required to determine if a particular number is in a list of 2n sorted numbers. Alternative: For any sorted list of n numbers, at most log2 n + 1 comparisons are required to determine if a particular number is in the list. Binary Search Complexity: O (log2 n)

Theorem 2.9 – Merge sorted lists Let A & B be sorted lists of numbers (both decreasing or increasing). Assume positive integer n represents the combined total of numbers in the 2 lists. A & B can be merged into a single list of sorted numbers in at most n-1 comparisons. Merge Complexity: O(n)

Theorem 1.3 – Number of subsets Yes, from chapter 1 If n is any non-negative number, the a set with n elements has exactly 2n subsets. Note – this can be used to determine the number of relations on a set.

Theorem 2.10 – Subsets of n items from r items Let S be a set of n elements & r is integer 0 <= r <= n. The number of subsets of S containing exactly r elements is n! / ( r! (n-r)! ) Application: How many 2 person committees can be selected from a set of 5 people?

Theorem 2.11 – Integers & Primes Every integer greater than 1 is either prime or a product of primes.

Homework – Section 2.6 Pages 91-92 – Exercises 1-8, 13 – 28 Homework – Supplementary Section Pages 95 – 96 – Exercises 1-8, 13-18, 31, 34, 35