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