Ⅰ introduction to set theory 1. sets and subsets representation of set:
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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 AB, and B A Properties Theorems, examples, and exercises. - PowerPoint PPT PresentationTRANSCRIPT
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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 AB, and B A Properties Theorems, examples, and exercises
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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
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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
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7.Functions one to one, onto, one-to-one correspondence Composite functions and Inverse
functions Cardinality, 0. Theorems, examples, and exercises
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II Combinatorics 1. Pigeonhole principle Pigeon and pigeonholes example , exercise
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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
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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
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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
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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
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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
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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
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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
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1.Let G be a tree with two or more vertices. Then G is a bipartite graph.
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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
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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
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5. Planar Graphs Euler’s formula, Corollary By Euler formula , Corollary, prove Example,exercise Vertex colorings Region(face) colorings Edge colorings Chromatic polynomials
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IV Abstract algebra 1. algebraic system n-ary operation: SnS function algebraic system : nonempty set S, Q1,
…,Qk(k1), [S;Q1,…,Qk] 。 Associative law, Commutative law, Identity
element, Inverse element, Distributive laws homomorphism, isomorphism Prove theorem 6.3 by theorem 6.3 prove
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2. Semigroup, monoid, group Order of an element order of group cyclic group Prove theorem 6.14 Example,exercise
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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
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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
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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?
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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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答疑 1 月 5 日上午 9:00-11:30 下午 1:00-3:30 地点 : 软件楼 3 楼办公室