COEN 231 Introduction to Discrete Mathematics FA L L 2 0 1 8 P R O F. J U N YA N Propositional Logics Definitions • Proposition/Statement, Counter-example • Not, and, or, xor, if-then, if-and-only-if • Converse, inverse, contrapositive • Tautology, contradiction, contingency, satisfiability • Logical equivalences for statements, conditionals, & bi-conditionals • Predicates, quantifiers, nested quantifiers, negating quantifiers • Arguments, valid arguments, and rules of inference • Direct proof, proof by contraposition, proof by contradiction, proof of equivalence, proof by cases, exhaustive proof, existence proof Calculations • Truth tables, logical equivalences, and proofs • English language to/from propositional logic 2 Set Theory Definitions • Set, element, membership, set-builder’s notation, and Venn diagram • Subset, proper subset, power set, and cardinality • Ordered 𝑛-tuple and Cartesian product • Union, intersection, disjoint, difference, and complement Calculations • Set operations • Set cardinality • Set identities Related • Counting • Relations 3 Functions Definitions • Function, domain, codomain, image, range, mapping, subset image • One-to-one, onto, one-to-one correspondence, inverse, composition Calculations • Determine properties of functions • Calculate floors and ceilings of values • Find the composition of functions Related • Sets • Sequences 4 Sequences Definitions • Sequence (as a function), geometric and arithmetic progression • Recurrence relations Calculations • Solution to recurrence relations • Summation of sequences Related • Sets • Counting 5 Matrices Definitions • Matrix, entry, square, matrix sum and product • Identity matrix, power, transpose, and symmetry of matrix • Zero-one matrices, Boolean operators, meet and join Calculations • Matrix sum and product • Meet and join Related • Relations • Graphs 6 Binaries Definitions • Positive integer representation and conversion • Positive fraction representation and conversion • Negative value representation, 1’s and 2’s complements, 2’s complement representation • Floating point representation Calculations • Binary representation, conversion, and complement • Binary addition and multiplication • Conversion to floating point notations 7 Algorithms Definitions • Algorithm, input, output, variables, pseudocode, indentation • Assign, for, while, if-then, return Calculations • Sort, search, greedy Related • Functions • Binaries • Graphs 8 Mathematical Inductions Definitions • Induction, basis step, inductive step • Conjecture for proof by induction Calculations • Proofs for summations, inequalities, divisibility, and set properties Related: • Proofs • Sets • Sequences 9 Counting Definitions • Sum rule and product rule • Inclusion-exclusion principle, tree diagram and pigeonhole principle • Permutations and combinations • Binominal coefficients Calculations • Counting the numbers related to all four definitions above Related: • Sets • Binaries • Relations • Graphs and trees 10 Relations Definitions • Relation, relation on a set, four properties of relations • Composition of relation, representation of relations • Equivalence relation and class, equivalent element, partition • Partial orderings, comparable element, Hess diagram, maximal/minimal and greatest/least elements, upper/lower bounds Calculations • Determine properties of a relation, equivalence, or poset • Representing relations • Finding/counting elements Related: • Sets, counting, graphs 11 Graphs Definitions • Graphs, types of graphs, adjacency, incidence, and graph matrices • Complete, bipartite, and isomorphic graphs • Paths, connectivity, Euler and Hamilton circuits • Trees, subtrees, rooted trees, spanning trees Calculations • Degrees of graphs • Finding bipartite graphs • Counting vertices and edges on graphs and trees Related: • Sets, counting, graphs, algorithms 12