Foundations of Mathematics: Sets, Relations, Functions, Combinatorics, Graphs, and Algebra

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 • Theorems, examples, and exercises

3. 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

4. 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 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 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

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. 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. 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. IVAbstract 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 relationon 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. 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. 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). Prove: • (1)f -1(S) an ideal ofR, where f -1(S)={xR|f (x)S} • (2)R/f -1(S) f (R)/S

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