MATH 139 W12 Review 1 Checklist 1. Exam Checklist. 1. Introduction to Predicates and Quantified Statements (chapters ).

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1 MATH 139 W12 Review 1 Checklist 1 Exam Checklist 1. Introduction to Predicates and Quantified Statements (chapters ). universal and existential statements truth set negations of universal and existential statements. contrapositive, converse and inverse of a universal statement. statements with multiple quantifiers forming negations of statements with multiple quantifiers. translating statements with multiple quantifiers into English and the reverse. Tarski world examples arguments with quantified statements universal instantiation. universal modus ponens. universal modus tollens. valid argument. using Venn diagrams to show validity or invalidity of an argument. converse error, inverse error. 2. Number Theory and Proofs (chapters ) prime number, composite number, rational number, irrational number. divisibility of numbers, when does one number divide another. proving an existential statement proof by example. proving a universal statement proof by exhaustion. disproof by counterexample. method of direct proof(eg. proof that the sum of two rational numbers is rational) properties of integers every integer n > 1 is divisible by a prime. every integer n > 1 can be factored into primes, n = p e 1 1 p e 2 2 p e k k. the quotient-remainder theorem: n = qd + r, 0 r < d. div and mod. what is an even number? odd number? the product of any two odd integers is odd the floor x and ceiling x functions. proofs by contradiction and contrapositive.

2 MATH 139 W12 Review 1 Checklist 2 proof that the sum of a rational and an irrational number is irrational. proof of the irrationality of 2 (and 3, 5, etc). proof that there are infinitely many primes. 3. Mathematical Induction (chapters ) Base step and inductive step in mathematical induction. proofs involving sums, eg n = n(n+1) 2. sum of a geometric series: 1 + r + r r n = rn+1 1 r 1. proofs involving divisibility, eg n 1 for all integers n 0. strong mathematical induction what is the difference in the assumption made in the inductive step? proofs involving sequences, eg. given b 1, b 2, b 3,... where b 1 = 4, b 2 = 12, and b k = b k 2 + b k 1 prove that 4 bn for all n = 1, 2, 3, Correctness of Algorithms (chapter 5.5) pre-conditions and post-conditions for an algorithm. loop invariants Guard, pre- and post-conditions for a loop. The Loop Invariant Theorem Basis property Inductive property Eventual falsity of the guard Correctness of the post-condition. Using the loop invariant theorem to show that certain simple algorithms are correct. 5. Functions (chapters ). domain, codomain, range, inverse image f 1 (y). arrow diagrams some basic functions: identity function i X, logarithmic function log a (x), Hamming distance function. One-to-one functions (injective), onto functions (surjective), bijective functions. the inverse function f 1 (x), how to find the inverse of a bijective function (eg. f(x) = 3x + 2). 6. Composition of Functions (chapter 7.3). Definition of g f. Various properties, such as f 1 f(x) = x and f f 1 (y) = y.

3 MATH 139 W12 Review 1 Checklist 3 If f : X Y and g : Y Z are both one-to-one functions, then g f is also one-to-one. If f : X Y and g : Y Z are both onto functions, then g f is also onto. If f and g are bijections, then g f is also a bijection. 7. Cardinality of Sets (chapter 7.4) How does one define sets of equal cardinality in terms of bijections? If the cardinality of a finite set is n, what does this mean in terms of bijections? Countable sets. Examples of countable infinite sets one should know: the set of integers, the set of rational numbers, the set of prime numbers. Uncountable sets eg. the set of real numbers. 8. Relations (chapter 8.1). What is a (binary) relation R for a pair of sets A and B. Arrow diagrams for a relation between two sets. Why is a function f : A B also a relation between A and B? When is a relation a function? The inverse R 1 of a relation R. The directed graph for a relation on a set. 9. Reflexivity, Symmetry, Transitivity, and Equivalence relations (chapters ) Certain properties of relations: reflexive. symmetric. transitive. How to test for reflexivity, symmetry, and transitivity by looking at the directed graph of a relation on a set. The transitive closure R t of a relation R. Equivalence relations relations which are reflexive, symmetric, and transitive. equivalence classes. certain basic equivalence relations, for example, the equivalence relation R defined on the set of integers where xry if x y mod d (where d is some fixed integer). How to find equivalence classes by looking at the directed graph of an equivalence relation on a set. 10. Graphs (chapter 10.1). vertex, edge, endpoints of an edge.

4 MATH 139 W12 Review 1 Checklist 4 loops, parallel edges, adjacent vertices, adjacent edges, isolated vertices. simple graph, complete graph, bipartite graph. subgraph of a graph. degree of a vertex, deg(v). total degree of a graph. Handshake Theorem total degree = twice the number of edges. Some basic corollaries eg. (i) total degree of a graph is always even (ii) number of odd vertices is even. 11. Trails, Paths and Circuits (chapter 10.2). walk, closed walk, path, simple path, circuit, simple circuit. connected graph. components of a graph, bridge of a graph. Euler circuit. Eulers Theorem a graph has an Euler circuit if and only if it is connected and all its vertices have even degree. Fleury s Algorithm for finding an Euler circuit. Hamilton circuit. 12. Matrix Representation Graphs (chapter 10.3) Adjacency matrix of a graph. Reconstructing a graph with help of its adjacency matrix. Counting walks using A 2, A 3, etc. 13. Trees (chapter ). What is a tree? Trivial tree. leaf (or terminal vertex) of a tree. internal (or branch) vertex of a tree. Theorem Every tree has at least 2 vertices of degree one. Theorem If a tree has n vertices, then it has n 1 edges. Rooted tree, root vertex, level of a vertex, height of a rooted tree. binary tree, full binary tree. Theorem If T is a full binary tree with k internal vertices, then T has k + 1 leaves. Theorem If T is a full binary tree with height h, then T has at most 2 h leaves. 14. Spanning Trees (chapter 10.7).

5 MATH 139 W12 Review 1 Checklist 5 Spanning tree of a graph. Weighted graph Minimum weight spanning tree of a weighted graph. Kruskal s Algorithm for finding a min. weight spanning tree. 15. Languages, Regular Expressions, and Automata (chapters ). Languages alphabet Σ, string over Σ. null string ϵ. language over Σ. operations on strings concatenation of two strings. concatenation of two languages. union of two languages. the Kleene closure L of a language L. regular expression over an alphabet Σ. How does a regular expression represent a language? Given a language, find a regular expression representing it. Finite state automata input alphabet states of an automaton, initial state S 0, accepting states. next-state function, next-state table. transition diagram when does an automaton accept an input string? language accepted by an automaton. Given an automaton, find the language accepted by it. Given a language, find an automaton which accepts it. Constructing an automaton which accepts the language represented by a regular expression. Kleene s theorem. 16. Real-Valued Functions (chapters ) real-valued function, graph of a real-valued function. power functions: x, x 2, x 3, x 4,.... graphs of basic power functions. multiple M f of a function f. increasing function, decreasing function. Big O, Ω notation.

6 MATH 139 W12 Review 1 Checklist 6 f(x) is order at most g(x): ie. f(x) is O(g(x)). g(x) is order at least f(x): ie. g(x) is Ω(f(x)). Given functions f(x) and g(x), show that f(x) is O(g(x)), or show that f(x) is Ω(g(x)).