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2.6 Equivalence Relation

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.

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2.6 Equivalence Relation

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  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 Ai and Aj are distinct elements of , then Ai∩Aj=. • 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 ={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

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

  20. (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.

  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：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.

  22. A2={2,3,6,12,24,36},(A2,|) • 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: A2={2,3,6,12,24,36},(A2,|) • 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,

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