1. 2.6 Equivalence Relation §1.Equivalence relation §Definition 2.18: A relation R on a set A is called an equivalence relation if it is reflexive, symmetric, and transitive. §Example: Let m be a positive integer with m>1. Show that congruence modulo m is an equivalence relation. R={(a,b)|a  b mod m} §Proof: (1)reflexive (for any a  Z ， aRa?) §(2)symmetric (for any aRb ， bRa?) §(3)transitive (for aRb ， bRc ， aRc?)

2. §2.Equivalence classes §partition §Definition 2.19: A partition or quotient set of a nonempty set A is a collection  of nonempty subsets of A such that §(1)Each element of A belongs to one of the sets in . §(2)If A i and A j are distinct elements of , then A i ∩A j = . §The sets in  are called the bocks or cells of the partition. §Example: Let A={a,b,c}, §P={{a,b},{c}},S={{a},{b},{c}},T={{a,b,c}}, §U={{a},{c}},V={{a,b},{b,c}},W={{a,b},{a,c},{c}}, §infinite

3. §Example ： congruence modulo 2 is an equivalence relation. §For any x  Z, or x=0 mod 2,or x=1 mod 2, i.e or x  E,or x  O. §And E∩O=  §E and O ， §{E, O} is a partition of Z

4. §Definition 2.20: Let R be an equivalence relation on a set A. The set of all element that are related to an element a of A is called the equivalence class of a. The equivalence class of a with respect to R is denoted by [a] R, When only one relation is under consideration, we will delete the subscript R and write [a] for this equivalence class. §Example ： equivalence classes of congruence modulo 2 are [0] and [1] 。 §[0]={…,-4,-2,0,2,4,…}=[2]=[4]=[-2]=[-4]=… §[1]={…,-3,-1,1,3,…}=[3]=[-1]=[-3]=… §the partition of Z  =Z/R={[0],[1]}

5. §Example: equivalence classes of congruence modulo n are ： §[0]={…,-2n,-n,0,n,2n,…} §[1]={…,-2n+1,-n+1,1,n+1,2n+1,…} §…§… §[n-1]={…,-n-1,-1,n-1,2n-1,3n-1,…} §A partition or quotient set of Z, §Z/R={[0],[1],…,[n-1]}

6. §Theorem 2.11 ： Let R be an equivalence relation on A. Then §(1)For any a  A, a  [a]; §(2)If a R b, then [a]=[b]; §(3)For a,b  A, If (a,b)  R, then [a]∩[b]=  ; §Proof ： (1)For any a  A ， aRa? §(2)For a,b  A, aRb, [a]  ?[b] ， [b]  ?[a] §For any x  [a],x  ?[b] when aRb ， i.e. x R b §for any x  [b], x  ?[a] when aRb ， i,.e.xRa §(3)For a,b  A, If (a,b)  R, then [a]∩[b]=  §Reduction to absurdity §Suppose [a]∩[b]≠ , Then there exists x  [a]∩[b]. §(4)

7. §The equivalence classes of an equivalence relation on a set form a partition of the set. §Equivalence relation  partition §Example ： Let A={1,2,3,4}, and let R={(1,1),(2,2),(3,3),(4,4), (1,3),(2,4),(3,1),(4,2)} is an equivalence relation. §Then the equivalence classes are:

8. §Conversely, every partition on a set can be used to form an equivalence relation. §Let  ={A 1,A 2,…,A n } be a partition of a nonempty set A. Let R be a relation on A, and aRb if only if there exists A i  s.t. a,b  A i. §i.e. R=(A 1  A 1 ) ∪ (A 2  A 2 ) ∪ … ∪ (A n  A n ) §R is an equivalence relation on A §Theorem 2.12 ： Given a partition {A i |i  Z} of the set A, there is an equivalence relation R that has the set A i, i  Z, as its equivalence classes

9. §Example: Let  ={{a,b},{c}} be a partition of A={a,b,c}. §Equivalence relation R=?

10. 2.7 Partial order relations §1.Partially ordered sets §Definition 2.21: A relation R on a set A is called a partial order if R is reflexive, antisymmetric, and transitive. The set A together with the partial order R is called a partially ordered set, or simply a poset, and we will denote this poset by (A,R). And the notation a ≼ b denoteds that (a,b)  R. Note that the symbol ≼ is used to denote the relation in any poset, not just the “lessthan or equals” relation. The notation a ≺ b denotes that a ≼ b but a  b.

11. §The relation ≦ on R ； §The relation | on Z + ； the relation  on P(A) 。 §partial order ， §(R, ≦ ), (Z +,/), (P(A),  ) are partially ordered sets 。 §Example ： Let A={1,2},P(A)={ ,{1},{2},{1,2}}, the relation on the powerset of A: ={( ,  ),( ,{1}),( ,{2}),( ,{1,2}), ({1},{1}),({1},{1,2}),({2},{2}),({2},{1,2}),({1,2},{1,2})}

12. §Example: Show that the inclusion relation  is a partial order on the power set of a set A §Proof:Reflexive: for any X  P(A), X  X. §Antisymmetric: For any X,Y  P(A), if X  Y and Y  X, then X=Y §Transitive: For any X,Y, and Z  P(A), if X  Y and Y  Z, then X  Z?

13. §The relation < on Z is not a partial order, since it is not reflexive §  and  is related, §{1} and {1,2} is related, §{2} and {1,2} is related ， but {1} and {2} is not related, incomparable §Related: comparable §not related: incomparable

14. §Definition 2.22 ： The elements a and b of a poset (A, ≼ ) are called comparable if either a ≼ b or b ≼ a. When a and b are elements of A such that neither a ≼ b nor b ≼ a, a and b are called incomparable.

15. § ≦ The relation ≦ on R ， § For any x,y  R, or x ≦ y, or y ≦ x ， §thus x and y is comparable §totally order

16. §Definition 2.23 ： If (A, ≼ ) is a poset and every elements of A are comparable, A is called a totally ordered or linearly ordered set, and ≼ is called a total order or linear order. A totally ordered set is also called a chain §The relation ≦ on Z is a total order. The relation | on Z is not a total order. The relation  on the power of a set A is not a total order.

17. §2 ． Hasse Diagrams §Hasse Diagrams §Digraph: predigestion §(1) partial order is reflexive ， §aRa ， §We shall delete all loop from the digraph §(2) Because a partial order is transitive ， We do not have to show those edges that must be present because of transitivity. §(3)If we assume that all edges are are pointed “upword”, we do not have to show the directions of the edges. §Hasse Diagrams

18. §The relation  on the power of a set A §P(A)={ ,{1},{2},{1,2}} §Example: A={2, 3, 6, 12, 24, 36}, (A, |) §A={1, 2, 3, 4, 5, 6},(A, ≦ )

19. §3.Extremal elements of partially ordered sets §Definition 2.24: Let (A, ≼ ) is a poset. An elements a  A is called a maximal element( 极大元 ) of A if there is no elements c in A such that a ≺ c. An elements b  A is called a minimal element ( 极小元 ) of A if there is no elements c in A such that c ≺ b. §Example ： A 1 ={1,2,3,4,5,6},(A 1,  ) §1 is a minimal element of A 1 §6 is a maximal element of A 1

20. §(A 1,|) §1 is a minimal element of A 1. §As these example shows, a poset can have more than one maximal element and more than one minimal element.

21. §Definition 2.25: Let (A, ≼ ) is a poset. An elements a  A is called a greatest element ( 最大元 ) of A if x ≼ a for all x  A. An elements a  A is called a least element ( 最小元 ) of A if a ≼ x for all x  A. §Note: difference between greatest element and maximal element §Example ： A 1 ={1,2,3,4,5,6},(A 1,  ) §1 is the least element of A 1. §6 is the greatest element of A 1 §(A 1,|) §1 is the least of A 1. §There is no greatest element.

22. §A 2 ={2,3,6,12,24,36},(A 2,|) §There is no greatest element. There is no least element.

23. §Definition 2.26: Let (A, ≼ ) is a poset, and B  A. An element a  A is called an upper bound ( 上界 ) of B if b ≼ a for all b  B. An element a  A is called a lower bound ( 下界 ) of B if a ≼ b for all b  B. §Example: A 2 ={2,3,6,12,24,36},(A 2,|) §P={2,3,6}, § all upper bounds of P are §P has no lower bounds.

24. §Definition 2.27: Let (A, ≼ ) is a poset, and B  A. An element a  A is called a least upper bound ( 最小上界 ) of B, (LUB(B)), if a is an upper bound of B and a ≼ a’, whenever a’ is an upper bound of B. An element a  A is called a greastest lower bound ( 最大下界 ) of B, (GLB(B)), if a is a lower bound of B and a’ ≼ a, whenever a’ is an lower bound of B.

25. §NEXT §Function P211 6.2 § P168, 5.1 §Exercise:P139 14,19,23, §P209 2,10,33 §P215 17,19,