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 AB, and B A Properties Theorems, examples, and exercises
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Ⅰ Introduction to Set Theory 1. Sets and Subsets Representation of set:
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 Presentation
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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
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
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
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 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),
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
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?
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