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# L08a

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### L08a

1. 1. Lecture 2 (part 1) Partial Orders Reading: Epp Chp 10.5
2. 2. Outline <ul><li>Revision </li></ul><ul><li>Definition of poset </li></ul><ul><li>Examples of posets </li></ul><ul><ul><li>In life </li></ul></ul><ul><ul><li>Finite posets </li></ul></ul><ul><ul><li>Infinite posets </li></ul></ul><ul><li>Notation </li></ul><ul><li>Visualization Tool: Hasse Diagram </li></ul><ul><li>Definitions </li></ul><ul><ul><li>maximal, greatest, minimal, least. </li></ul></ul><ul><ul><li>2 Theorems </li></ul></ul><ul><li>7. More Definitions </li></ul><ul><ul><li>Comparable, chain, total-order, well-order </li></ul></ul>
3. 3. 1. Revision Concrete World Abstract World Relation R from A to B a. ___ has been to ___ {John, Mary, Peter} {Tokyo, NY, HK} {(John,Tokyo), (John,NY), (Peter, NY)} b. ___ is in ___ {Tokyo, NY} {Japan, USA} {(Tokyo,Japan), (NY,USA)} c. ___ divides ___ {1,2,3,4} {10,11,12} {(1,10),(1,11),(1,12), (2,10), (2,12),(3,12), (4,12)} d. ___ less than ___ {1,2,3} {1,2,3} {(1,2),(1,3),(2,3)} ___ R ___ A B R  A  B Q: What can you do with relations? A: (1) Set Operations; (2) Complement; (3) Inverse; (4) Composition Q: What happens if A = B ?
4. 4. 1. Revision Concrete World a. ___ same age as ___ {John, Mary, Peter} {(John,John), (Mary,Mary) (Peter,Peter), (Mary,Peter), (Peter,Mary)} b. ___ same # of elements as ___ { {}, {1}, {2}, {3.4} } { ({},{}), ({1},{1}), ({2},{2}) ({3,4},{3,4}) ({1},{2}), ({2},{1}) c. ___  ___ { {}, {1}, {2}, {1,2} } { ({},{}), ({},{1}), ({},{2}), ({},{1,2}), ({1},{1}), ({1},{1,2}), ({2},{2}), ({2},{1,2}) ({1,2},{1,2}) } d. ___  ___ {1,2,3} {(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)} ___ R ___ A R  A 2 Relation R on A “ Everyone is related to himself” Reflexive “ If x is related to y and y is related to z , then x is related to z .” Transitive “ If x is related to y , then y is related to x ” Symmetric “ If x is related to y and y is related to x , then x = y .” Anti-Symmetric
5. 5. 1. Revision <ul><li>Given a relation R on a set A, </li></ul><ul><ul><li>R is reflexive iff </li></ul></ul><ul><ul><li> x  A, x R x </li></ul></ul><ul><ul><li>R is symmetric iff </li></ul></ul><ul><ul><li> x,y  A, x R y  y R x </li></ul></ul><ul><ul><li>R is anti-symmetric iff </li></ul></ul><ul><ul><li> x,y  A, x R y  y R x  x=y </li></ul></ul><ul><ul><li>R is transitive iff </li></ul></ul><ul><ul><li> x,y  A, x R y  y R z  x R z </li></ul></ul>
6. 6. 2. Definition <ul><li>Given a relation R on a set A, </li></ul><ul><ul><li>R is an equivalence relation iff </li></ul></ul><ul><ul><li>R is reflexive , symmetric and transitive . ( Last Lecture) </li></ul></ul><ul><ul><li>R is a partial order iff </li></ul></ul><ul><ul><li>R is reflexive , anti-symmetric and transitive . </li></ul></ul><ul><ul><li>(This Lecture) </li></ul></ul>
7. 7. 2. Definition <ul><li>Given a relation R on a set A, </li></ul><ul><ul><li>R is an partial order (or partially-ordered set; or poset) iff R is reflexive , anti-symmetric and transitive . </li></ul></ul><ul><li>Q: How do I check whether a relation is an partial order? </li></ul><ul><li>A: Just check whether it is reflexive, anti-symmetric and transitive. Always go back to the definition. </li></ul><ul><li>Q: How do I check whether a relation is reflexive, symmetric and transitive? </li></ul><ul><li>A: Go back to the definitions of reflexive, symmetric and transitive. </li></ul>
8. 8. 3.1 Examples (Partial Orders in life) <ul><li>PERT - Program Evaluation and Review Technique. </li></ul><ul><li>CPM - Critical Path Method </li></ul><ul><li>Used to deal with the complexities of scheduling individual activities needed to complete very large projects. </li></ul>Let T be the set of all tasks. We define a relation R on T such that x R y iff x = y or task x must be done before task y .
11. 11. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul>
12. 12. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q1: Is R reflexive ? </li></ul><ul><li>Reflexive :  x  A, x R x (Always go back to the definition) </li></ul><ul><li>Yes, R is reflexive. </li></ul>
13. 13. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q2: Is R anti-symmetric ? </li></ul><ul><li>Anti-symmetric : </li></ul><ul><li> x,y  A, x R y  y R x  x = y </li></ul><ul><li>(Again, the definition!) </li></ul>
14. 14. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), ( 3 , 1 ), ( 1 ,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q2: Is R anti-symmetric ? </li></ul><ul><li>Anti-symmetric : </li></ul><ul><li> x,y  A, x R y  y R x  x = y </li></ul><ul><li>(Again, the definition!) </li></ul>True Always false LHS: False
15. 15. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), ( 3 , 1 ), ( 1 ,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q2: Is R anti-symmetric ? </li></ul><ul><li>Anti-symmetric : </li></ul><ul><li> x,y  A, x R y  y R x  x = y </li></ul><ul><li>(Again, the definition!) </li></ul>LHS: False Vacuously/blankly/stupidly True
16. 16. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q2: Is R anti-symmetric ? </li></ul><ul><li>Anti-symmetric : </li></ul><ul><li> x,y  A, x R y  y R x  x = y </li></ul><ul><li>(Again, the definition!) </li></ul>LHS: False Vacuously True
17. 17. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q2: Is R anti-symmetric ? </li></ul><ul><li>Anti-symmetric : </li></ul><ul><li> x,y  A, x R y  y R x  x = y </li></ul><ul><li>(Again, the definition!) </li></ul>LHS: False Vacuously True
18. 18. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q2: Is R anti-symmetric ? </li></ul><ul><li>Anti-symmetric : </li></ul><ul><li> x,y  A, x R y  y R x  x = y </li></ul><ul><li>(Again, the definition!) </li></ul><ul><li>Carry on checking… </li></ul><ul><li>Yes, it’s anti-symmetric </li></ul>
19. 19. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q3: Is R transitive ? </li></ul><ul><li>Transitive : </li></ul><ul><li> x,y,z  A, x R y  y R z  x R z </li></ul><ul><li>(DEFINITION!!!) </li></ul>
20. 20. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {( 0 , 0 ), (3,1), (1,1), ( 0 , 4 ), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q3: Is R transitive ? </li></ul><ul><li>Transitive : </li></ul><ul><li> x,y,z  A, x R y  y R z  x R z </li></ul><ul><li>(DEFINITION!!!) </li></ul>True True True True
21. 21. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), ( 3 , 1 ), ( 1 , 1 ), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q3: Is R transitive ? </li></ul><ul><li>Transitive : </li></ul><ul><li> x,y,z  A, x R y  y R z  x R z </li></ul><ul><li>(DEFINITION!!!) </li></ul>True True True True
22. 22. 3.2 Examples (Finite Partial Orders) <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4)} </li></ul><ul><li>Is R a partial order? </li></ul><ul><li>Q3: Is R transitive ? </li></ul><ul><li>Transitive : </li></ul><ul><li> x,y,z  A, x R y  y R z  x R z </li></ul><ul><li>(DEFINITION!!!) </li></ul><ul><li>Carry on checking… </li></ul><ul><li>Yes, R is transitive. </li></ul>
23. 23. 3.3 Examples (Common Infinite Posets) <ul><li>Let R be a relation on Z, such that </li></ul><ul><li>x R y iff x  y </li></ul><ul><li>R is a partial order </li></ul><ul><li>Reflexive:  x  Z, x  x </li></ul><ul><li>Anti-symmetric:  x,y  Z, x  y  y  x  x=y </li></ul><ul><li>Transitive:  x,y,z  Z, x  y  y  z  x  z </li></ul>We will abbreviate the description of this relation to R = (Z,   )
24. 24. 3.4 Examples (Common Infinite Posets) <ul><li>Let R be a relation on Z + , such that </li></ul><ul><li>x R y iff x | y </li></ul><ul><li>R is a partial order </li></ul><ul><li>Reflexive:  x  Z + , x | x </li></ul><ul><li>Anti-symmetric:  x,y  Z + , x|y  y|x  x=y </li></ul><ul><li>Transitive:  x,y,z  Z + , x|y  y|z  x|z </li></ul>We will abbreviate the description of this relation to R = (Z + , |  )
25. 25. 3.5 Examples (Common Infinite Posets) <ul><li>Let R be a relation on P(A), such that </li></ul><ul><li>X R Y iff X  Y </li></ul><ul><li>R is a partial order </li></ul><ul><li>Reflexive :  X  P(A), X  X </li></ul><ul><li>Anti-symmetric :  X,Y  P(A), X  Y  Y  X  X=Y </li></ul><ul><li>Transitive :  X,Y,Z  P(A), X  Y  Y  Z  X  Z </li></ul>We will abbreviate the description of this relation to R = (P(A),  )
26. 26. 4. Notation <ul><li>In general, if we describe a partial order relation as: </li></ul><ul><li>Let R be a relation on A, such that </li></ul><ul><li>x R y iff x op y </li></ul><ul><li>… we will shorten the description to </li></ul><ul><li>R = (A, op ) </li></ul>Of course, this can be done only when the relation can be described in terms of a simple operator . We will not be able to this if the relation is described by a complicated logical expression
27. 27. 4. Notation <ul><li>In general, if we describe a partial order relation as: </li></ul><ul><li>Let R be a relation on A, such that </li></ul><ul><li>x R y iff x op y </li></ul><ul><li>… we will shorten the description to </li></ul><ul><li>R = (A, op ) </li></ul>Hence we have : 1. R = (Z,  ) 2. R = (Z + , | ) 3. R = (P(A),  )
28. 28. 4. Notation <ul><li>There are times when we discuss partial orders in general . In such cases we may write: </li></ul><ul><li>R = (A,  ) </li></ul><ul><li>as a general partial order. </li></ul><ul><li>We choose the ‘  ’ symbol to represent a general ordering operator because it looks like ‘  ’. </li></ul><ul><li>This is done due to the fact that the ordering of the elements in the set convey the idea of one below the other (something like  on Z). </li></ul>
29. 29. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4), (1,4)} </li></ul>0 3 1 4 2 Let’s simplify the diagram 1. Eliminate all reflexive loops. 0 3 1 4 2
30. 30. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4), (1,4)} </li></ul>0 3 1 4 2 Let’s simplify the diagram 2. Eliminate all transitive arrows. 0 3 1 4 2 0 3 1 4 2
31. 31. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4), (1,4)} </li></ul>0 3 1 4 2 Let’s simplify the diagram 3. (a) Draw all arrow heads pointing upwards, and (b) eliminate arrow heads. 0 3 1 4 2 0 3 1 4 2 0 3 1 4 2
32. 32. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4} </li></ul><ul><li>Let R = {(0,0), (3,1), (1,1), (0,4), (2,2), (3,4), (3,3), (4,4), (1,4)} </li></ul>0 3 1 4 2 The result is a Hasse Diagram . 0 3 1 4 2 0 3 1 4 2 0 3 1 4 2
33. 33. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4}. Let R = (A,   </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>1. Eliminate all reflexive loops. 0 4 2 1 3
34. 34. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4}. Let R = (A,   </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>0 4 2 1 3 1. Eliminate all reflexive loops. 2. Eliminate all transitive arrows.
35. 35. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4}. Let R = (A,   </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>0 4 2 1 3 1. Eliminate all reflexive loops. 2. Eliminate all transitive arrows.
36. 36. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4}. Let R = (A,   </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>0 4 2 1 3 1. Eliminate all reflexive loops. 2. Eliminate all transitive arrows. 3. (a) Draw all arrow heads pointing upwards, and (b) eliminate arrow heads.
37. 37. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {0,1,2,3,4}. Let R = (A,   </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>0 4 2 1 3 1. Eliminate all reflexive loops. 2. Eliminate all transitive arrows. 3. (a) Draw all arrow heads pointing upwards, and (b) eliminate arrow heads.
38. 38. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {1,2,3,…,10}. Let R = (A, |  </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>You may draw the Hasse Diagram immediately if you are able to. 1 2 3 5 7 10 4 8 9 6
39. 39. 5. Visualisation Tool: Hasse Diagram <ul><li>Let A = {1,2,3}. Let R = (P(A),   </li></ul><ul><li>Draw the Hasse Diagram. </li></ul>R = ( { {} , {1} , {2} , {3} , {1,2} , {1,3} , {2,3} , {1,2,3} } ,  ) {} {1} {3} {2} {1,2} {1,3} {1,2,3} {2,3}
40. 40. <ul><li>To be continued </li></ul>