CSE 20 Discrete Mathematics for Computer Science Prof

CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2 Final review

3 What did we study? Boolean logic: propositions, predicates, formulas, truth tables, CNFs, DNFs Data representation: graphs, sets, functions, relations, equivalence relations Proofs: direct, contrapositive, contradiction, cases, induction, strong induction Number theory: factorization to primes, GCD, basis representation, modular arithmetics Number theory algorithms: div-mod, fast exponentiation, casting out 9 s

4 What did we really study? Language of mathematics Expressing intuitive (or not so much) ideas in a formal language Giving formal proofs for mathematical theorems

5 Boolean logic

6 Propositional logic What is a proposition? How to translate from English to logic, and vice versa Connectives Truth tables CNFs, DNFs Equivalence rules

7 Propositional logic You should know: Translating English from/to logic Negating / De-Morgan laws Computing truth tables from formulas Computing CNFs / DNFs from truth table Equivalence rules, derivation rules

8 Predicate logic Universal / existential quantifiers Multiple quantifiers Negation / De-Morgan laws The importance of the domain

9 Predicate logic You should know: Translating English from/to logic Negating / De-Morgan laws Equivalence rules, derivation rules Deciding if a quantified formula is true/false in a specific domain

10 Data representation

11 Graphs What do graphs represent? Examples. . Terminology: Directed vertices, edges, degree / undirected graphs (Eulerian graphs – not for final)

12 Sets Definition, notations, examples Operations: union, intersection, complement, difference, power set, Cartesian product Expressing sets (eg primes) using set builder notation Venn diagrams, algebraic / symbolic proofs Set sizes, and how these change under the above operations Infinite sets

13 Relations What do they represent? Examples. . Connection Types: How to graphs reflexive, symmetric, transitive to prove / disprove that a relation is reflexive / symmetric / transitive

14 Equivalence relations Definition How to prove/disprove Important example: modular arithmetics (equivalent MOD m)

15 Functions Definition, Injective, examples surjective, bijective Inverse function Proving properties of set sizes using functions

16 Proofs

17 Direct proof To prove p q Assume p, plug in definitions, derive q

18 Contrapositive proof To prove p q Prove instead ~q ~p Assume ~q, plug in definitions, derive ~p

19 Proof by contradiction To prove p q Assume p, ~q and derive a contradiction (either to one of the assumptions, or to a basic axiom)

20 Proof by cases To prove p q Partition Prove assumption p into cases each case individually, using any of the previous proof techniques

21 Induction Useful to prove theorems of the form: for all n>=1, some P(n) holds Base: n=1 (say) Basic induction: assume true for n-1, prove for n Strong induction: assume true for all k=1, …, n 1, prove for n

22 Number theory + algorithms

23 Primes Definition of primes / composites Every integer has a unique factorization as a product of prime numbers If p is prime, p|xy p|x or p|y Hard to factor a number to primes – basis for RSA encryption

24 GCD Greatest Can common divisor of 2 numbers find efficiently using Euclid’s algorithm

25 Modular arithmetics Definitions “x MOD m” Addition, multiplication Inversion when m is prime

26 Number theory algorithms DIV-MOD: Divide a by b, get quotient q and reminder r: a=bp+r Fast exponentiation: compute ab using only ~log(b) multiplications Casting by 9 out 9 s: check if a number is divisible

27 Review questions

28 Question 1

29 Question 2

30 Question 3 Let G be an undirected graph with n vertices and no loops. What is the maximal possible number of edges in G? A. n B. n 2 C. n(n-1)/2 D. n 2/2

31 Question 4

32 Question 5

33 Question 6

34 Some words for conclusion I enjoyed teaching you, I hope that you enjoyed the class as well I hope that it opened your mind to the wonderful world of mathematics, and its many applications in computer science; this was just the first step Good luck in the final!