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

5. 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

Cardinality, 0. 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 generating functions and Exponential generating functions 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. Let the number of edges of G be m. Suppose m≥(n2-3n+6)/2, where n is the number of vertices of G. Show that (G-S)≤|S| for each nonempty proper subset S of V(G). Hamilton cycle! 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 1.Let G be a tree with two or more vertices. Then G is a bipartite graph. 2.Let G be a simple graph with n vertices. Show that ifδ(G) >[n/2]-1, then G is a tree or contains three spanning trees at least.

m-ary tree , full m-ary tree, optimal tree 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 Let G is a planar graph. If (G)=2 then G is a bipartite graph Let G is a planar graph. If (G)=2 then G does not contain any odd simple circuit.

Prove theorem 6.3 by theorem 6.3 prove 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

Let  is an equivalence relation on the group G, and if axax’ then x x‘ for a,x,x‘G. Let H={x|xe, xG}. Prove: H is a subgroup of G. xx-1=ex=xe xe, y e x-1xy=ye=x-1x

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? Let ring A there be one and only a right identity element. Prove A is an unitary ring.

Let e is right identity element of A. For aA，ea-a+eA， For xA，x(ea-a+e)=? ea-a+e right identity element of A ea-a+e＝e， ea=a， e is identity element of A.。

Quotient ring, Find zero-divisors, ideal, Integral domains? 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

1. Let f : R→S be a ring homomorphism, with a subring A of R 1. Let f : R→S be a ring homomorphism, with a subring A of R. Show that f(A) is a subring of S. 2. Let f: R→S be a ring homomorphism, with an ideal A of R. Does it follow that f(A) is an ideal of S? 3.Prove Theorem 6.36 Theorem 6.36: Let  be a ring homomorphism from ring [R;+,*] to ring [S;+’,*’]. Then (1)[(R);+’,*’] is a subring of [S;+’,*’] (2)[ker;+,*] is an ideal of [R;+,*].

4. Let f : R→T be a ring homomorphism, and S be an ideal of f (R) 4. Let f : R→T be a ring homomorphism, and S be an ideal of f (R). Prove: (1)f -1(S) an ideal of R, where f -1(S)={xR|f (x)S} (2)R/f -1(S) f (R)/S

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