Functions
DefinitionofFunction
Definition: Let A and B be nonempty sets. A
functionffromAtoB,whichisdenotedf:AB,is
a relation from A to B such that for all aA, f(a)
containsjustoneelementofB.
Aspecialkindofbinaryrelation
Under f,eachelementinthedomaino
Trees
Lecture 8 Discrete Mathematical Structures
Rooted Tree
Let A be a set, and let T be a relation on A. T is a rooted tree if there is a vertex v0 in A with the property that there exists a unique path in T from v0 to every other vertex in A, but no pa
Transportation Network
Lecture 10 Discrete Mathematical Structures
Transport Networks
The unique node with out-degree 0 The sink 4 1 5 4 2 2 3 2 3 3 3 Capacity of edge, Ci,j 5 4 6
The unique node with in-degree 0 The source It is assume that all edges are
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Discrete Mathematical Structures
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Discrete Mathematic
Course description
f 9 f 9 f 9
f 9 =Icfw_ - c 9 / Fs =
.R[ 9
04/01
Algebraic Systems and Groups
Lecture 13 Discrete Mathematical Structures
Algebraic Operations
Function :AnB is called an n-nary operation from A to B.
Binary operation: :A AB (:A AA) An example: a new operation * defined on the set of real number, using
Relations and Digraphs
Lecture 5 Discrete Mathematical Structures
Relations and Digraphs
Cartesian Product Relations Matrix of Relation Digraph Paths in Digraph
Ordered Pair and Cartesian Product
Ordered pairs:
(a,b) (a,b) = (c,d) iff a = c and b = d
Poset and Lattice
Lecture 7 Discrete Mathematical Structures
P P P
Partial Order
Reflexive, antisymmetric and transitive
Generalization of less than or equal to
Denotation: Example 1: set containment
Note: not any two of sets are comparable
Exampl
Logic
The discipline that deals with the methods of reasoning
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The example of reason
If I am your teacher then I should give you lessons I am your teacher So I should give you lessons Is it right? Maybe just now!
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Another example of r
Group
Lecture 14 Discrete Mathematic Structures
Group
Group axioms
Association Identity Inverse property
Example
Addition group on integers (Z,+) All one-to-one functions on cfw_1,2,3, plus composition of function: S3
Inverse Property of a System
For
Graphs
Lecture 9 Discrete Mathematical Structures
A C B D
Problem of Crossing River
Problem: A person(P), a wolf(W), a lamb(L) and a cabbage(C) will cross a river by a boat which can carry any two of them once. Wolf and lamb, or, lamb and cabbage, cannot
Equivalence
Lecture 6 Discrete Mathematical Structures
Equivalence
Properties of Relation
Reflexivity Symmetry Transitivity
Equivalence Equivalence and Partition
Reflexivity
Relation R on A is
Reflexive if for all aA, (a,a)R Irreflexive if for all a
Counting
Counting
Countable Set Permutations and combinations Pigeonhole Principle Recurrence Relations
Countable Set
A set A is countable if and only if we can arrange all of its elements in a linear list in a definite order.
Definite means that we ca