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
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