binary relations

1.

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2. give examples of relations with specified properties, S, not R, not T

example: inequality

3. equivalence relations

Reflexive; Symmetric; Transitive

4. Suppose R ⊆ S ×S is an equivalence relation, the equivalence class [s] (w.r.t. R) of an element s ∈ S is

[s] = {t : t ∈ S and sRt}

! ! !  s R t if and only if [s] = [t]

prove: [s]=[t]

[s] ⊆ [t]

x∈[s]

so s R x

so x R s   (by S)

so x R t   (by T)

so t R x   (by S)

so x∈[t]

we can do the same with x∈[t]

 

5. partitions

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举例:show that m~n iff m^2=n^2(mod 5) is an equivalence on S={1,2...7}, find all the equivalence classes

[1]={1,4,6}=[4]=[6]

[2]={2,3,7}=[3]=[7]

[5]={5}

6. partial order

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7. Using Hasse diagram to represent finite poset:

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8. ordering concepts

a) Minimal and maximal elements (they always exist in every finite poset)

b) Minimum and maximum — unique minimal and maximal element (might not exist)

e.x.  若1向上指向2,3,2与3上面均没有更大的数,故两者均为maximal,但不是maximum,1既为minimal也是minimum

c) binary relations_第5张图片

 

d)Lattice — poset where lub(x,y) and glb(x,y) exist for every pair of elements x,y

同样上面的例子,因为2,3没有lub, 所以is not a lattice

9. Total orders

A total order is a partial order that also satisfies:
(L) Linearity (any two elements are comparable):
For all x,y either: x ≤ y or y ≤ x (or both if x = y)

在有限集合上,所有的总阶都是“同构的”             

在一个有限的集合里有很多种可能性

10. 

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11.

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12. combining orders

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13. 类比字典排序

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e.x. Let B = {0,1} with the usual order 0 < 1. List the elements 101,010,11,000,10,0010,1000 of B∗ in the

(a) Lexicographic order 000,0010,010,10,1000,101,11

(b) Lenlex order(by length first) 10,11,000,010,101,0010,1000

14.  If a set Σ is totally ordered, then the corresponding lexicographic partial order on Σ∗ also must be totally ordered

answer: True

If a set Σ is totally ordered, then the corresponding lenlex order on Σ∗ also must be totally ordered.

answer: True

Every finite partially ordered set has a Hasse diagram

answer: True

Every finite partially ordered set has a topological sorting

answer: True

Every finite partially ordered set has a minimum element

answer: False

Every finite totally ordered set has a maximum element

answer: True

An infinite partially ordered set cannot have a maximum element

answer: False

 

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