 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  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 generating functions and Exponential generating functions  e x =1+x+x 2 /2!+…+x n /n!+…;  x+x 2 /2!+…+x n /n!+…=e x -1;  e -x =1-x+x 2 /2!+…+(-1) n x n /n!+…;  1+x 2 /2!+…+x 2n /(2n)!+…=(e x +e -x )/2;  x+x 3 /3!+…+x 2n+1 /(2n+1)!+…=(e x -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  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: S n  S function  algebraic system ： nonempty set S, Q 1,…,Q k (k  1), [S;Q 1,…,Q k ] 。  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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