1. 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

2. 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

3. 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

4. Cardinality, 0. 7.Functions one to one, onto,
one-to-one correspondence
Composite functions and Inverse functions
Cardinality, 0.
Theorems, examples, and exercises

5. II Combinatorics 1. Pigeonhole principle Pigeon and pigeonholes
example，exercise

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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.

14. 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

15. 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

16. 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.

17. 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

18. 2. Semigroup, monoid, group
Order of an element
order of group
cyclic group
Prove theorem 6.14
Example,exercise

19. 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

20. 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

21. 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

22. 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.

23. 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.。

24. 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

25. 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;+,*].

26. 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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