Example: Let Ii-a, b,,c be a partition of A=(a,b, c). Equivalence relation r-
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 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 denoted that (a, bER. Note that the symbol s 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
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 ab
The relation on R: The relation on Z: the relation c on P(A) partial order, R, s ),(Z, /),(P(A),C are partially ordered sets Example:LetA={1,2},P(A)={,{1}2{2},{1,2},the relation on the powerset of A: ={(,),(,{1}),(,{2}),(,{1,2}) (1},{1}),({1},{1,2})({2},{2})2({(2},{1,2}),({1,2},{1,2})}
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 c is a partial order on the power set of a set a Proof: Reflexive: for any X∈P(△),XcX Antisymmetric: For any X,Y EP(A), if XcY and Yax. then Xy Transitive: For any X,Y, and ZEP(A), if XcY and Ycl, then Xcz?
Example: Show that the inclusion relation is a partial order on the power set of a set A Proof:Reflexive: for any XP(A), XX. Antisymmetric: For any X,Y P(A), if XY and YX, then X=Y Transitive: For any X,Y, and ZP(A), if XY and YZ, then XZ?
The relation on Z is not a partial order. since it is not reflexive o and o is related 1 and ( 1, 2) is related, 12 and ( 1, 2 is related, but 1 and (2 is not related, incomparable Related: comparable not related: incomparable
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