Transcript dmi29.ppt

 ⅠIntroduction to Set Theory
 1. Sets and Subsets
 Representation of set:
 Listing elements, Set builder notion, Recursive
definition
 , , 
 P(A)
 2. Operations on Sets
 Operations and their Properties
 A=?B
 AB, and B A
 Properties
 Theorems, examples, and exercises
 3. Relations and Properties of relations
 reflexive ,irreflexive
 symmetric , asymmetric ,antisymmetric
 Transitive
 Closures of Relations
 r(R),s(R),t(R)=?
 Theorems, examples, and exercises
 4. Operations on Relations
 Inverse relation, Composition
 Theorems, examples, and exercises
 5. Equivalence Relations
 Equivalence Relations
 equivalence class
 6.Partial order relations and Hasse Diagrams
 Extremal elements of partially ordered sets:
 maximal element, minimal element
 greatest element, least element
 upper bound, lower bound
 least upper bound, greatest lower bound
 Theorems, examples, and exercises
 7.Functions
 one to one, onto,
 one-to-one correspondence
 Composite functions and Inverse
functions
Cardinality, 0.
 Theorems, examples, and exercises
II Combinatorics
 1. Pigeonhole principle
 Pigeon and pigeonholes
 example，exercise
 2. Permutations and Combinations
 Permutations of sets, Combinations of sets
 circular permutation
 Permutations
and Combinations of
multisets
 Formulae
 inclusion-exclusion principle
 generating functions
 integral solutions of the equation
 example，exercise
 Applications of Inclusion-Exclusion principle
 theorem 3.15,theorem 3.16,example,exercise
 Applications
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generating functions
Exponential generating functions
and
ex=1+x+x2/2!+…+xn/n!+…;
x+x2/2!+…+xn/n!+…=ex-1;
e-x=1-x+x2/2!+…+(-1)nxn/n!+…;
1+x2/2!+…+x2n/(2n)!+…=(ex+e-x)/2;
x+x3/3!+…+x2n+1/(2n+1)!+…=(ex-e-x)/2;
 3. recurrence relation
 Using Characteristic roots to solve recurrence
relations
 Using Generating functions to solve recurrence
relations
 example，exercise
III Graphs
 1. Graph terminology
 The degree of a vertex，(G),  (G),
Theorem 5.1 5.2
 k-regular, spanning subgraph, induced
subgraph by V'V
 the complement of a graph G,
 connected, connected components
 strongly connected, connected directed
weakly connected
2. connected, Euler and Hamilton
paths
 Prove: G is connected
 (1)there is a path from any vertex to any
other vertex
 (2)Suppose G is disconnected
 1) k connected components(k>1)
 2)There exist u,v such that is no path
between u,v
 Prove
that the complement of a
disconnected graph is connected.
 Let G be a simple graph with n vertices.
Show that ifδ(G) >[n/2]-1, then G is
connected.
 Show that a simple graph G with an
vertices is connected if it has more than
(n-1)(n-2)/2 edges.
 Theorems, examples, and exercises
 Determine whether there is a Euler cycle
or path, determine whether there is a
Hamilton cycle or path. Give an
argument for your answer.
 Find the length of a shortest path
between a and z in the given weighted
graph
 Theorems, examples, and exercises
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3.Trees
Theorem 5.14
spanning tree minimum spanning tree
Theorem 5.16
Example: Let G be a simple graph with n
vertices. Show that ifδ(G) >[n/2]-1, then G has
a spanning tree
 First: G is connected，
 Second:By theorem 5.16⇒ G has a spanning
tree
 Path ,leave
 1.Let G be a tree with two or more
vertices. Then G is a bipartite graph.
 Find a minimum spanning tree by
Prim’s algorithms or Kruskal’s
algorithm
 m-ary tree , full m-ary tree, optimal
tree
 By Huffman algorithm, find optimal
tree , w(T)
 Theorems, examples, and exercises
 4. Transport Networks and Graph
Matching
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Maximum flow algorithm
Prove:theorem 5.24, examples, and exercises
matching, maximum matching.
M-saturated, M-unsaturated
perfect matching
(bipartite graph), complete matching
M-alternating path (cycle)
M-augmenting path
Prove:Theorem 5.25
Prove: G has a complete matching,by Hall’s
theorem
 examples, and exercises
5. Planar Graphs
 Euler’s formula, Corollary
 By Euler formula，Corollary, prove
 Example,exercise
 Vertex colorings
 Region(face) colorings
 Edge colorings
 Chromatic polynomials
 IV Abstract algebra
 1. algebraic system
 n-ary operation: SnS function
 algebraic system ：nonempty set S,
Q1,…,Qk(k1), [S;Q1,…,Qk]。
 Associative law, Commutative law, Identity
element, Inverse element, Distributive laws
 homomorphism, isomorphism
 Prove theorem 6.3
 by theorem 6.3 prove
2. Semigroup, monoid, group
 Order of an element
 order of group
 cyclic group
 Prove theorem 6.14
 Example,exercise
 3. Subgroups, normal subgroups ,coset,
and quotient groups
 By theorem 6.20(Lagrange's Theorem), prove
 Example: Let G be a finite group and let the
order of a in G be n. Then n| |G|.
 Example: Let G be a finite group and |G|=p. If
p is prime, then G is a cyclic group.
 Let G =, and consider the binary operation. Is
[G; ●] a group?
 Let G be a group. H=. Is H a subgroup of G?
 Is H a normal subgroup?
 Proper subgroup
 4. The fundamental theorem of
homomorphism for groups
 Homomorphism kernel
 homomorphism image
 Prove: Theorem 6.23
 By
the
fundamental
theorem
of
homomorphism
for
groups,
prove¨[G/H;][G';]
 Prove: Theorem 6.25
 examples, and exercises
 5. Ring and Field
 Ring, Integral domains, division rings,
field
 Identity of ring and zero of ring
commutative ring
 Zero-divisors
 Find zero-divisors
 Let R=, and consider two binary
operations. Is [G; +,●] a ring, Integral
domains, division rings, field?
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characteristic of a ring
prove: Theorem 6.32
subring, ideal, Principle ideas
Let R be a ring. I=…
Is I a subring of R?
Is I an ideal?
Proper ideal
 Quotient ring, Find zero-divisors, ideal, Integral
domains?
 By the fundamental theorem of homomorphism for
rings(T 6.37), prove [R/ker;,] [(R);+’,*’]
 examples, and exercises
答疑
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