A Brief Summary for Exam 2

1
A Brief Summary for Exam 2

• Subject Topics
• Number theory (sections 2.4 - 2.5)
• Prime numbers
• Definition
• Relative prime
• Fundamental theorem of arithmetic (unique prime
  factorization of integers)
• Division
• Definition (a b iff b ac) and properties
• Division algorithm a dq r
• Modula operation (a mod d)
• gcd and lcm
• Their definitions
• gcd(a, b)lcm(a, b) ab (why?)
• Euclidean algorithm for gcd (both iterative and
  recursive

2

• Mathematical Induction Recursion (sections 3.1
  - 3.5)
• Sequence and summation
• Definition of sequence (ordered list of elements)
• Summation notations (lower/upper limits, double
  summation)
• Useful sequences (arithmetic, geometric,
  Fibonacci, etc.) and their summation formulas
• Induction
• Rationale and relation to natural numbers
• Two parts of the proof
• basis step, inductive step (start with inductive
  hypothesis)
• Strong induction
• Structural induction

3

• Recursion
• Basic idea of recursion
• Recursive definition of
• Sequences, functions, sets
• Two parts base case and recursion
• Relations to induction
• prove recursively defined properties by
  induction
• Recursive algorithms
• Pros and cons (wrt iterative algorithms)

4

• Counting (sections 4.1 4.5)
• Useful rules
• Sum rule (disjoint) tasks done at same time
• A1 ? A2 A1 A2
• Product rule (disjoint) tasks done at different
  time
• A1 ? A2 A1 A2
• Inclusion exclusion rule (overlapping) tasks
  done at same time A1 ? A2 A1 A2 - A1 ?
  A2
• Pigeonhole Principle
• Idea and rationale
• at least one box containing at least ?N/k? of the
  objects.

5

• Permutations and combinations
• Definitions of permutations, r-permutations,
  r-combinations
• Formulae for (P(n,n), P(n, r), and C(n, r)
• P(n, r) n!/(n r)!
• C(n, r) n!/r!(n-r)!
• Relationship between permutation and combinations
• P(n, r) C(n, r)P(r, r)
• Pascal triangle and Binomial coefficients

6

• Discrete Probability (sections 5.1 - 5.2)
• Experiments, outcomes, and sample space
• Use counting techniques to determine sample space
• p(s) for each s?S
• 0 ? p(s) ? 1 for each s?S
• ?s?S p(s) 1
• If all outcome are equally probable, then p(s)
  1/S
• Events and event probability
• E ? S, P(E) ?s?E p(s) (P(E) E/S if
  outcomes are equally likely)
• Use counting techniques to determine samples in E
• Complementary event P(E) 1 P(-E).
• Help with Venn diagram

7

• Conditional probability P(EF)
• Definition probability of E, given F
  (probability of E or in subspace F? S)
• Relation to joint probability
• p(EF) p(E?F)/p(F) or p(EF) E?F/F
• p(E?F) p(EF)P(F) p(FE)P(E)
• Inclusion-exclusion rule
• p(E?F) p(E) p(F) p(E?F)
• Bayes theorem
• p(FE) p(EF)p(F)/p(E)

8

• Independence
• Events E and F are independent of each other if
• p(EF) p(E) (Es probability not depending on
  F)
• P(E?F) p(E) p(F) p(E)p(F)
• Bernoulli Trials
• Experiment with two outcomes, s and f,
• p P(s), q P(f) 1 p (because p q 1)
• n independent trials with k s (and n k f)
• C(n, k)pkqn-k

9

• 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
• No questions will be outside of this summary and
  lecture notes