8.6 partial orderings
DESCRIPTION
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 } - PowerPoint PPT PresentationTRANSCRIPT
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8.6 Partial Orderings
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DefinitionPartial 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 }
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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 iff aRb≼• Also a b iff a≺ ≺ b and a≠b
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Examples and non-examples of posets (S, ≼)
• 1. (Z, ≤) proof
• 2. (Z, ≥)
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More examples
• 3. (Z, |) where | is “divides”
• 4. ( Z+ , |)
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…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
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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.”
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Comparable, incomparable elements• For each set, find
comparable elements incomparable (if any):
1. (Z, ≤ ) using the usual ≤ 2. (Z+, |)
3. (P(S), ) where S={1,2,3}
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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.
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Are these examples total orders or not?
• (Z, ≤ )
• (Z+, |)
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Lexicographic Order (dictionary)
Things to consider:Longer lengths or different lengths in words Ex: Discreet<discreteDiscreet<discreetnessDiscrete<discretion
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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 ≺ 1 b1 or (a1=b1 and a2 ≺ 2 b2)
• Letter or number examples
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(A1xA2, ≼) is a poset
• Proof Method?• Proof – see book
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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)
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Ex. 1. S={1,2,3,4}; poset (S, ≤)
Original digraph reduced diagram4|3|2|1
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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
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Ex 3: Hasse diagram of (P({a,b,c}), )
•
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Ex. 4: Hasse of ({2,4,5,10,12,20,25,}, | )
•
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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.
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greatest element
• Claim: The greatest element, when it exists, is unique.
• Proof:– Method?
• Similarly, the least element, when it exists, is unique.
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Upper bound,…
• Def: Let (S, ≼) be a poset and A S.– If uS and a ≼ u for all aA,u is 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.
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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 Bb c
lower bounds of Ba
• find lub and glb
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Ex. 6: A={4,6,8} with “divides” relation
lub(A)=glb(A)=
Note: lub=?glb=?
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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+, ≤)
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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.
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({2,4,5,10,12,20,25}, | )Recall Hasse diagram for ({2,4,5,10,12,20,25}, | )
Create several topological sorts.
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House Ex- book
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Advising example
•