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 = a*c) and properties Division algorithm: a = dq + r Modula operation (a mod d) –gcd and lcm Their definitions gcd(a, b)*lcm(a, b) = a*b (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 |A 1  A 2 | = |A 1 | + |A 2 | Product rule: (disjoint) tasks done at different time |A 1  A 2 | = |A 1 | * |A 2 | Inclusion – exclusion rule: (overlapping) tasks done at same time |A 1  A 2 | = |A 1 | + |A 2 | - |A 1  A 2 | –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(E|F) Definition: probability of E, given F (probability of E or in subspace F  S ) Relation to joint probability –p(E|F) = p(E  F)/p(F) or p(E|F) = |E  F|/|F| –p(E  F) = p(E|F)P(F) = p(F|E)P(E) –Inclusion-exclusion rule: p(E  F) = p(E) + p(F) – p(E  F) –Bayes’ theorem p(F|E) = p(E|F)p(F)/p(E)

8. –Independence Events E and F are independent of each other if p(E|F) = p(E) (E’s 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)p k q n-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