BITS Pilani Pilani Campus Discrete Structures for Computer Science (CS F222/IS F222) SK Hafizul Islam, Ph.D [email protected]
BITS PilaniPilani Campus
BITS PilaniPilani Campus
Discrete Structures for Computer Science(CS F222/IS F222)
SK Hafizul Islam, Ph.D
BITS Pilani, Pilani Campus
• Consider the renovation of a hostel. In this process several tasks were undertaken
– Remove Asbestos
– Replace windows
– Paint walls
– Refinish floors
– Assign offices
– Move in office furniture
Introduction
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• Clearly, some things had to be done before others could begin– Asbestos had to be removed before anything (except assigning offices)
– Painting walls had to be done before refinishing floors to avoid ruining them, etc.
• On the other hand, several things could be done concurrently:– Painting could be done while replacing the windows
– Assigning offices could be done at anytime before moving in office furniture
• This scenario can be nicely modeled using partial orderings
Introduction
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• Definitions:
– A relation R on a set A is called a partial order if it is
• Reflexive [for any a A, (a, a) R]
• Antisymmetric [For a, bA, if (a, b)R and (b, a)R, then a = b]
• Transitive [For a, b, cA, if (a, b)R and (b, c)R, then (a, c)R]
– The set A together with a partial order relation R is called a partially ordered set (POSET).
Partial Order
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• Partial orderings are used to give an order to setsthat may not have a natural one
• In our renovation example, we could define anordering such that (a, b)R if “a must be donebefore b can be done”.
Partial Order
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• A POSET is denoted by (A p).
• We use the notation:
– apb, when (a, b)R and ab [a strictly precedes b]
• The notation p is not “less than”.
• The notation p is used to denote any partial ordering
POSET
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• The relation “less than or equal to” defined over aset of real numbers is a POSET, i.e., (R, ) is a POSET.
• Which of the following sets with the given relationsfrom POSETs?
– (Z, ≥)
– (Z, >)
– (P, ), where P = {, {a}, {b}, {a, b}}
– (P, )
– (Z, |), where I is the divisibility relation
– (Z+, I)
POSET
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
Cover:
• Let (A, p) be a POSET and a, bA. The element b iscalled the cover of a if apb and no cA exists suchthat apcpb.
– b is the cover of a if b is the immediate successor of a.
• Example: A = {2, 3, 6, 12, 18, 36} with the partialorder relation “divides”.– 2p6, 6 is the cover of 2, but 3 is not the cover of 2.
– 36 is not a cover of 6, since 6p12p36 or 6p18p36.
Hasses Diagram
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
Covering relation:
• For a finite poset (A, p), we can find the cover of
every element in A.
• The set of pairs (a, b) such that “b covers a” is calledthe covering relation of (A, p).
• Example: A = {2, 3, 6, 12, 18, 36} with the partialorder relation “divides”.– The covering relation of (A, p) = {(2, 6), (3, 6), (6, 12), (6,
18), (12, 36), (18, 36)}.
Hasses Diagram
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• The Hasses diagram of a finite poset (A, p) is a graph
in which
– The elements are represented as vertices
– If b is a cover of a, then this relation is shown by placing bhigher than a and providing a edge between them.
Hasses Diagram
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CS F222/IS F222 8 September 2015
BITS Pilani, Pilani Campus
• A = {2, 3, 6, 12, 18, 36} with the partial order relation“divides”.– The covering relation of (A, p) = {(2, 6), (3, 6), (6, 12), (6, 18),
(12, 36), (18, 36)}.
Hasses Diagram
36
12 18
6
2 3
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CS F222/IS F222 8 September 2015