1 / 28

8.6 Partial Orderings

8.6 Partial Orderings. Definition. Partial ordering – a relation R on a set S that is Reflexive, Antisymmetric , and Transitive Examples? R={( a,b )| a is a subset of b } R={( a,b )| a divides b } on {1,2,3,4} R={(1,1),(1,2),(1,3),(1,4),(2,2),…} R={( a,b )| a≤ b }

oceana
Download Presentation

8.6 Partial Orderings

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 8.6 Partial Orderings

  2. Definition Partial ordering– a relation R on a set S that is Reflexive, Antisymmetric, and Transitive Examples? • R={(a,b)| a is a subset of b } • R={(a,b)| a divides b } on {1,2,3,4} • R={(1,1),(1,2),(1,3),(1,4),(2,2),…} • R={(a,b)| a≤ b } • R={(a,b)| a=b+1 }

  3. Partially ordered set (poset) • (S,R) -- a set S and a relation R on S, that is R, A, and T. • Often we use (S, ≼) • Note: ≼ is a generic symbol for R • It includes the usual ≤, but it is more general. It also covers other poset relations: divides, subset,… • We say a ≼ b iffaRb • Also a≺biff a≺b and a≠b

  4. Examples and non-examples of posets (S, ≼) • 1. (Z, ≤) proof • 2. (Z, ≥)

  5. More examples • 3. (Z, |) where | is “divides” • 4. ( Z+ , |)

  6. …examples • 5. (P(S), ) where S={1,2,3} and P(S) is the power set • 6. (P(S), ) where S is a set and P(S) is the power set

  7. Comparable • Def: The elements a and b of a poset (S, ≼) are said to be “comparable” if either a ≼b or b ≼a. • Otherwise, they are “incomparable.”

  8. Comparable, incomparable elements • For each set, find

  9. totally (linearly) ordered set • Def: • A poset (S, ≼) is a totally (linearly) ordered set if every two elements of S are comparable. • ≼ is then a total order, and S is a chain.

  10. Are these examples total orders or not? • (Z, ≤ ) • (Z+, |)

  11. Lexicographic Order (dictionary) Things to consider: Longer lengths or different lengths in words Ex: Discreet<discrete Discreet<discreetness Discrete<discretion

  12. Lexicographic order • Suppose (A1, ≼1) and (A2, ≼2) are two posets. • Let (a1, a2), (b1, b2) A1xA2 • Let (a1, a2) ≺(b1, b2) in case either a1≺ 1b1 or (a1=b1 and a2≺2 b2) • Letter or number examples

  13. (A1xA2, ≼) is a poset • Proof Method? • Proof – see book

  14. Hasse diagram • Hasse diagram—a diagram that contains sufficient information to find a partial ordering • Algorithm: • create a digraph with directed edges pointing up • remove all loops (reflexive is assumed) • remove any (a,c) where (a,b) and (b,c) are present (transitivity assumed) • remove arrows (direction up is assumed)

  15. Ex. 1. S={1,2,3,4}; poset (S, ≤) Original digraph reduced diagram 4 | 3 | 2 | 1

  16. Ex. 2: (S, ≼) where S={1,2,3,4,6,8,12} and ≼ ={(a,b)|a divides b} Shorthand: ({1,2,3,4,6,8,12}, | ) 8 12 | | 4 6 | | 2 3 | 1

  17. Ex 3: Hassediagram of (P({a,b,c}), )

  18. Ex. 4: Hasse of ({2,4,5,10,12,20,25,}, | )

  19. Maximal, minimal… • Def: • Let (S, ≼) be a poset and a S. • a is maximal in (S, ≼) if there does not exist b S such that a≺b. • a is minimal in (S, ≼) if there does not exist b S such that b ≺a. • a is the greatest element of (S, ≼) if b ≼a for all b S. • a is the least element of (S, ≼) if a ≼b for all b S. • Find examples of maximal, greatest elements,… in above examples.

  20. greatest element • Claim: The greatest element, when it exists, is unique. • Proof: • Method? • Similarly, the least element, when it exists, is unique.

  21. Upper bound,… • Def: Let (S, ≼) be a poset and AS. • If uSand a ≼ u for all aA,uis an upper bound of A. • If l S and l ≼ a for all a A, l is an lower bound of A. • x is a least upper bound of A , lub(A), if x is an upper bound and x ≼ z for every upper bound z of A. • y is a greatest lower bound of A , glb(A), if y is a lower bound and z ≼ y for every lower bound z of A. • Remark: lub and glb are unique when they exist.

  22. Ex. 5 (S, ≼ ) A={b,d,g}, B=(d,e} h i upper bounds of A: | lub(A)= g f lower bounds of A: | | glb(A)= d e | | upper bounds of B b c lower bounds of B a • find lub and glb

  23. Ex. 6: A={4,6,8} with “divides” relation lub(A)= glb(A)= Note: lub=? glb=?

  24. Well-ordered set Def: (S, ≼) is well-ordered set if it is a poset such that ≼ is a total ordering and every nonempty subset of S has a least element. Find Ex and non-ex.: • (Z+, ≤) • (Z, ≤) • (Z+ x Z+, lexicographic order) • (R+, ≤)

  25. Topological sorting Use: for project ordering Def: A total ordering ≼ is compatible with the partial order R if a ≼ b whenever aRb. The construction of such a total order is called a topological sorting. Lemma: Every finite non-empty poset (S, ≼ ) has a minimal element.

  26. ({2,4,5,10,12,20,25}, | ) Recall Hassediagram for ({2,4,5,10,12,20,25}, | ) Create several topological sorts.

  27. House Ex- book

  28. Advising example

More Related