?Introduction to Set Theory

1

• ?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

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
• ex1xx2/2!xn/n!
• xx2/2!xn/n!ex-1
• e-x1-xx2/2!(-1)nxn/n!
• 1x2/2!x2n/(2n)!(exe-x)/2
• xx3/3!x2n1/(2n1)!(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(kgt1)
• 2)There exist u,v such that is no path between
  u,v
• Shortest-path problem

10

• Prove that the complement of a disconnected graph
  is connected.
• Let G be a simple graph with n vertices. Show
  that ifd(G) gtn/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(n2-3n6)/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!
• Theorems, examples, and exercises

12

• 3.Trees
• Theorem 5.12
• spanning tree minimum spanning tree
• Theorem 5.14
• Example Let G be a simple graph with n vertices.
  Show that ifd(G) gtn/2-1, then G has a spanning
  tree
• First G is connected,
• SecondBy theorem 5.14? 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 ifd(G) gtn/2-1, then G is a tree or
  contains three spanning trees at least.

14

• Find a minimum spanning tree by Prims algorithms
  or Kruskals 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
• Provetheorem 5.22, examples, and exercises
• matching, maximum matching.
• M-saturated, M-unsaturated
• perfect matching
• (bipartite graph), complete matching
• M-alternating path (cycle)
• M-augmenting path
• ProveTheorem 5.23
• Prove G has a complete matching,by Halls
  theorem
• examples, and exercises

16

• 5. Planar Graphs
• Eulers 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

• IV Abstract algebra
• 1. algebraic system
• n-ary operation Sn?S function
• algebraic system nonempty set S, Q1,,Qk(k?1),
  SQ1,,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 Gp. 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
  Hxx?e, x?G. Prove H is a subgroup of G.
• xx-1e?xxe
• x?e, y ?e
• x-1xyy?ex-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, proveG/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-ae?A,
• For ?x?A,x(ea-ae)?
• ea-ae right identity element of A
• ea-aee,
• eaa,
• e is left 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

• Example Let R be a commutative ring, and H be an
  ideal of R. Prove that quotient ring R/H is an
  integral domain ? For any a,b?R, if ab?H, then
  a?H or b?H.
• Proof (1)If quotient ring R/H is an integral
  domain, then a?H or b?H when ab?H where a,b?R.
• (2)R is a commutative ring, and H be an ideal of
  R. If a?H or b?H when ab?H where a,b?R, then
  quotient ring R/H is an integral domain.

26

• ??
• 1? ???900-1100
• ??130-400
• ?? ??? ??
• 1? ???130-400
• ?? ???