A Brief Summary for Exam 2

1. A Brief Summary for Exam 2 Subject Topics • Mathematical Induction & Recursion (sections 3.1 - 3.5) • Sequence and summation • Definitions (lower/upper limits, double summation) • Useful sequences and their summations (arithmetic, geometric, Fibonacci) • Induction • Definition and relation to natural number • Three parts of the proof basis step, inductive hypothesis, inductive step • Strong induction

2. Recursion • Basic idea of recursion • Recursive definition of • Sequences, functions, sets • Two parts: base case and recursion • Relations to induction • Recursive algorithms • Pros and cons (wrt iterative algorithms)

3. Counting (sections 4.1 – 4.5) • Useful rules: • Sum rule: disjoint, done at different time |A1  A2| = |A1| + |A2| • Product rule: disjoint, done at same time |A1  A2| = |A1| * |A2| • Inclusion – exclusion rule: overlapping, done at different time |A1  A2| = |A1| + |A2| - |A1  A2| • Pigeonhole Principle • Idea and rationale • at least one box containing at least N/k of the objects.

4. Permutations and combinations • Definitions of permutations, r-permutations, r-combinations • Relationship between permutation and combinations • Formulae for (P(n,n), P(n, r), and C(n, r) • Pascal triangle and Binomial Coefficients

5. Recurrence Relations (sections 6.1 and 6.2) • Definition of recurrence relation and its solution • Relationship with recursive definition • Ideas of modeling with recurrence relations • Ideas of solving linear homogeneous recurrence relation

6. Types of Questions • Conceptual • Definitions of terms • True/false • Multiple choice • Problem solving • Work with small concrete example problems • Proofs • Simple theorems or propositions • Especially proof by mathematical induction