7.11 A proof involving composition Variation in terminology... 88

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1 Contents Preface xi 1 Math review Some sets Pairs of reals Exponentials and logs Some handy functions Summations Strings Variation in notation Logic A bit about style Propositions Complex propositions Implication Converse, contrapositive, biconditional Complex statements Logical Equivalence i

2 ii 2.8 Some useful logical equivalences Negating propositions Predicates and Variables Other quantifiers Notation Useful notation Notation for 2D points Negating statements with quantifiers Binding and scope Variations in Notation Proofs Proving a universal statement Another example of direct proof involving odd and even Direct proof outline Proving existential statements Disproving a universal statement Disproving an existential statement Recap of proof methods Direct proof: example with two variables Another example with two variables Proof by cases Rephrasing claims Proof by contrapositive Another example of proof by contrapositive

3 iii 4 Number Theory Factors and multiples Direct proof with divisibility Stay in the Set Prime numbers GCD and LCM The division algorithm Euclidean algorithm Pseudocode A recursive version of gcd Congruence mod k Proofs with congruence mod k Equivalence classes Wider perspective on equivalence Variation in Terminology Sets Sets Things to be careful about Cardinality, inclusion Vacuous truth Set operations Set identities Size of set union Product rule Combining these basic rules

4 iv 5.10 Proving facts about set inclusion An abstract example An example with products A proof using sets and contrapositive Variation in notation Relations Relations Properties of relations: reflexive Symmetric and antisymmetric Transitive Types of relations Proving that a relation is an equivalence relation Proving antisymmetry Functions and onto Functions When are functions equal? What isn t a function? Images and Onto Why are some functions not onto? Negating onto Nested quantifiers Proving that a function is onto A 2D example Composing two functions

5 v 7.11 A proof involving composition Variation in terminology Functions and one-to-one One-to-one Bijections Pigeonhole Principle Permutations Further applications of permutations Proving that a function is one-to-one Composition and one-to-one Strictly increasing functions are one-to-one Making this proof more succinct Variation in terminology Graphs Graphs Degrees Complete graphs Cycle graphs and wheels Isomorphism Subgraphs Walks, paths, and cycles Connectivity Distances Euler circuits

6 vi 9.11 Bipartite graphs Variation in terminology way Bounding Marker Making Pigeonhole point placement Graph coloring Why care about graph coloring? Proving set equality Variation in terminology Induction Introduction to induction An Example Why is this legit? Building an inductive proof Another example Some comments about style A geometrical example Graph coloring Postage example Nim Prime factorization Variation in notation Recursive Definition Recursive definitions

7 vii 12.2 Finding closed forms Divide and conquer Hypercubes Proofs with recursive definitions Inductive definition and strong induction Variation in notation Trees Why trees? Defining trees m-ary trees Height vs number of nodes Context-free grammars Recursion trees Another recursion tree example Tree induction Heap example Proof using grammar trees Variation in terminology Big-O Running times of programs Asymptotic relationships Ordering primitive functions The dominant term method Big-O

8 viii 14.6 Applying the definition of big-o Proving a primitive function relationship Variation in notation Algorithms Introduction Basic data structures Nested loops Merging two lists A reachability algorithm Binary search Mergesort Tower of Hanoi Multiplying big integers NP Finding parse trees What is NP? Circuit SAT What is NP complete? Variation in notation Proof by Contradiction The method is irrational There are infinitely many prime numbers Lossless compression

9 ix 17.5 Philosophy Collections of Sets Sets containing sets Powersets and set-valued functions Partitions Combinations Applying the combinations formula Combinations with repetition Identities for binomial coefficients Binomial Theorem Variation in notation State Diagrams Introduction Wolf-goat-cabbage puzzle Phone lattices Representing functions Transition functions Shared states Counting states Variation in notation Countability The rationals and the reals Completeness Cardinality

10 x 20.4 Cantor Schroeder Bernstein Theorem More countably infinite sets P(N) isn t countable More uncountability results Uncomputability Variation in notation Planar Graphs Planar graphs Faces Trees Proof of Euler s formula Some corollaries of Euler s formula K 3,3 is not planar Kuratowski s Theorem Coloring planar graphs Application: Platonic solids A Jargon 245 A.1 Strange technical terms A.2 Odd uses of normal words A.3 Constructions A.4 Unexpectedly normal B Acknowledgements and Supplementary Readings 251 C Where did it go? 255

MATH 363: Discrete Mathematics

MATH 363: Discrete Mathematics Learning Objectives by topic The levels of learning for this class are classified as follows. 1. Basic Knowledge: To recall and memorize - Assess by direct questions. The