1. Review: Discrete Mathematics and Its Applications

2. Contents
Mathematical Logic
Set Theory
Counting
Graph and Tree

3. Propositional Logic Propositions and Logical Operators
Propositional Formula and Its Classification
Tautology
Contradiction
Contingence

4. How to determine whether a compound proposition is a tautology (contradiction, contingence)?
− truth table
− logical equivalence
− normal forms
Propositional Equivalences
− Propositions logically equivalent is an equivalent relation
− Some important equivalences

5. Propositional Normal Forms
− Conjunctive Clauses and Disjunctive Normal Form
− Miniterm and Full Disjunctive Norm
− Conjunctive Normal Form
Valid of Argument
− Valid of Argument
− Rules of Inference

6. Predicate Logic Predicates and Quantifiers
universe of discourse, banding
Logical Equivalence
− De Morgan’s laws for predicates
− Quantifiers—-handle with care!

7. Inference for Quantified Statements
But
Still holds
Still holds
Inference for Quantified Statements

8. Set Theory Set and Set operations − Subsets, Proper Subsets
− Set Operations and Set Identities
Power Set

9. Cartesian Products
ordered pair
Cardinality of Finite sets
− Principle of Inclusion-exclusion

10. Infinite Set
Cardinality Infinite Set
− cardinality
− a infinite set has a proper subset with the same cardinality
Countable Infinite Set
− has the same cardinality with
− Properties of the countable sets:
The union of two countable sets, finite number of countable sets,
countable number of countable sets are countable.
− Some special infinite sets:

11. Uncountable Infinite Set
− Let countable infinite set, uncountable
− uncountable

12. Relation
Relations and their representing −
− Representing methods: Set, Matrix, Graph
Properties of relations
− Reflexive, Symmetric, Transitive
Combining relations
− Composite, Inverse

13. Closures of relations
be a relation on set .
− reflexive closure , where
− symmetric closure
− transitive closure
Equivalence relation
− Equivalence relation: reflexive, symmetric, transitive
− Equivalence classes and their properties
− Partition and equivalence relation

14. Partial ordering
− Partial order: reflexive, antisymmetric and transitive. A set
together with a partial ordering is called a partial order set
(denoted by ), totally order
− Hasse Diagram
− maximal element, minimal element, greatest element, least element, upper bound, lower bound, least upper bound, greatest lower bound
− Lattice: A partially ordered set in which pair of elements has both a least upper bound and a greatest lower bound

15. Counting Basic Counting Two basic counting principle − the sum rule
− the product rule
The Pigeonhole Principle
Permutations and combinations
− without repetition and with repetitions
Generating permutations and combinations

16. Advanced Counting Techniques
− Recurrence relations
− Generating function
− Solving recurrence relations、Counting with Generating function
− Solving linear (non-)homogeneous recurrence relations with
constant
Characteristic Equation, Characteristic roots

17. Graphs and Trees Introduction to Graphs
− Types of graphs: undirected [simple, multigraph, pseudograph],
directed [directed graph, directed multigraph]
− Some special simple graphs
− Representing graphs and graph isomorphism
− Connectedness
Euler graph and Hamilton graph
Planar Graph and Coloring
− Euler formula, Chromatic number
Shortest Path
− Dijkstra’s algorithm

18. Tree
Definition and Properties of Trees
− tree: connected + no simple circuit
− unique simple path between any two of its vertices
− tree: vertices, edges
− A full m-ary tree: internal vertices, vertices
− If an m-ary tree of height has leaves, then If
the m-ary trees is full and balanced, then
Applications of Trees
− Binary search trees
− Prefix codes

19. Spanning Tree
− Algorithms for constructing spanning trees: depth-first search
(backtracking), breadth-first search.
Minimal Spanning Tree
− Algorithms for constructing minimal spanning trees: Kruskal’s
algorithm, Prim’s algorithm