Ⅰ introduction to set theory 1. sets and subsets representation of set: listing elements, set...

Ⅰ Introduction 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 AB, 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 generating functions and

Exponential generating functions 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

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

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: SnS function algebraic system ： nonempty set S, Q1,

…,Qk(k1), [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?

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

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