Definition 2. 20: Let R be an equivalence relation on a set A 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]}

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]}

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)

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:

Conversely, every partition on a set can be used to form an equivalence relation. Let ={A1,A2,…,An} be a partition of a nonempty set A. Let R be a relation on A, and aRb if only if there exists Ai s.t. a,bAi. i.e. R=(A1A1)∪(A2A2)∪…∪(AnAn) R is an equivalence relation on A Theorem 2.12：Given a partition {Ai|iZ} of the set A, there is an equivalence relation R that has the set Ai, iZ, as its equivalence classes

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

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.

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})}

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?

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

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.

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

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.

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.

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,≦)

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：A1={1,2,3,4,5,6},(A1,) 1 is a minimal element of A1 6 is a maximal element of A1

(A1,|) 1 is a minimal element of A1. As these example shows, a poset can have more than one maximal element and more than one minimal element.

Definition 2. 25: Let (A, ≼) is a poset 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：A1={1,2,3,4,5,6},(A1,) 1 is the least element of A1. 6 is the greatest element of A1 (A1,|) 1 is the least of A1. There is no greatest element.

A2={2,3,6,12,24,36},(A2,|) There is no greatest element. There is no least element.

Definition 2. 26: Let (A, ≼) is a poset, and BA 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: A2={2,3,6,12,24,36},(A2,|) P={2,3,6}, all upper bounds of P are P has no lower bounds.

Definition 2. 27: Let (A,≼) is a poset, and BA 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.

RA×B,R is a relation from A to B，DomRA。 (a,b)R (a, c)R (a,b)R (a, c)R unless b=c function DomR=A， (everywhere)function。 22

Chapter 3 Functions 3.1 Introduction Definition3.1: Let A and B be nonempty sets. A relation is an everywhere function from A to B, denoted by f : AB, if for every aA, there is one and only b B so that (a,b) f, we say that b=f (a). The set A is called the domain of the function f. If XA, then f(X)={f(a)|aX} is called the image of X. The image of A itself is called the range of f, we write Rf. If YB, then f -1(Y)={a|f(a)Y} is called the preimage of Y. A function f : AB is called a mapping. If (a,b) f so that b= f (a), then we say that the element a is mapped to the element b. 23

(2)if (a,b) and (a,b')f, then b=b‘ Relation: (a,b),(a,b')R, Everywhere function: (1)Domf=A； (2)if (a,b) and (a,b')f, then b=b‘ Relation: (a,b),(a,b')R, function : if (a,b) and (a,b')f, then b=b‘ Relation: DomRA Everywhere function: DomR=A 24

Example：Let A={1,2,3,4},B={a,b,c}, R1={(1,a),(2,b),(3,c)}, R2={(1,a),(1,b),(2,b),(3,c),(4,c)}, R3={(1,a),(2,b),(3,b),(4,a)} Example: Let A ={-2,-1, 0,1,2} and B={0,1,2,3,4,5}. Let f={(-2,0),(-1,1), (0,0),(1,3),(2,5)}. f is an everywhere function. X={-2,0,1}, f(X)=? Y={0,5}, f -1(Y)=? 25

(1)If A1A2, then f(A1) f(A2) (2) f(A1∩A2) f(A1)∩f(A2) Theorem 3.1: Let f be an everywhere function from A to B, and A1 and A2 be subsets of A. Then (1)If A1A2, then f(A1) f(A2) (2) f(A1∩A2) f(A1)∩f(A2) (3) f(A1∪A2)= f(A1)∪f(A2) (4) f(A1)- f(A2) f(A1-A2) Proof: (3)(a) f(A1)∪f (A2) f(A1∪A2) (b) f(A1∪A2) f(A1)∪f (A2) 26

(4) f (A1)- f (A2) f (A1-A2) for any y f (A1)-f (A2) 27

Exercise:P226 2,10, 11,33 P232 17,19,23,26,27,28 P188 2