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Partial Orders Definitions A relation R on set A is a partial order if ... - PowerPoint PPT Presentation


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Partial Orders. Definitions. A relation R on set A is a partial order if it is: reflexive antisymmetric transitive. A is called a partially ordered set or poset . [A;R] means A is partially ordered by R. Example: [ P({1,2});  ] ( draw its digraph ). Definitions .

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definitions
Definitions
  • A relation R on set A is a partial order if it is:
    • reflexive
    • antisymmetric
    • transitive.
  • A is called a partially ordered set or poset.
  • [A;R] means A is partially ordered by R.
  • Example: [ P({1,2});  ] (draw its digraph).
definitions1
Definitions ...
  • is the prototype of a partial order
  • a, bA are comparable under  if either ab or ba.
  • Otherwise, they are said to be incomparable
  • If a, bA are comparable, then [A; ] is totally ordered and A is a chain.
examples
Examples
  • [ Z;  ],
    • where Z is the set of integers
    •  has the usual meaning.
    • Z is totally ordered.
  • [ Z+; | ],
    • where Z+ is a set of positive integers
    • | is the (evenly) divides operator (e.g., 2 | 10 )
hasse diagrams
Hasse Diagrams
  • A Hasse diagram is a directed graph (digraph), where
    • self-loops are omitted
    • arcs implied by transitivity are omitted.
  • Let Dn denote the set of positive divisors of n.
  • Draw Hasse diagram for [ D8; | ]
  • Draw Hasse diagram for [ D6; | ]
hasse diagrams1
Hasse Diagrams
  • What n yields a poset [Dn; | ] whose Hasse diagram is a cube of dimension:
    • 0
    • 1
    • 2
    • 3
examples1
Examples
  • Let S be a set. [ P(S);  ] is a poset.
  • Draw the Hasse diagram for:
    • S = 
    • S = {1}
    • S = {1, 2}
    • S = {1, 2, 3}
    • What do you think the Hasse diagram looks like for S = {1, 2, 3, 4}? For S = {1, . . ., n}?
computer science example
Computer Science Example
  • Consider this sequence of Java assignments:

a = b + d + c;

d = a*(b + d) + c;

e = (b + d)*c;

(Draw an operator graph for these statements.)

    • Sequence these operations in a way that is compatible with their partial order.
    • Where are the longest paths?
topological sorting
Topological Sorting
  • Let G = (V,E) be a directed graph where
    • v V represents a task;
    • (u, v)  E means that task u must be completed before v can be started.
  • G cannot have cycles.
  • Problem: Find a schedule for G that respects the partial order.
composing relations
Composing Relations
  • Let R be a relation from A to B.
  • Let S be a relation from B to C.
  • The composition of R & S, denoted RS, is

RS = { xRSy | zB, xRz  zSy }.

  • Example:
    • R = { (1,2), (3,4), (2,4) }
    • S = { (2,4), (2,3), (4,1) }
    • RS = { (1,4), (1,3), (3,1), (2,1) }
the transitive closure
The Transitive Closure
  • Let R be a relation from S to S.
  • RR is usually denoted R2.
  • Ri+1 = RiR.
  • The transitive closure, R+, of a binary relation R is:
reflexive transitive closure
Reflexive-Transitive Closure
  • R0 = { (x, x) | x  S }
  • The reflexive-transitive closure, denoted R*, is: R* = R0  R+. That is,
example
Example

R

a

b

c

d

e

R0

Self-loops not shown

a

b

c

d

e

R+

a

b

c

d

e

R*

Self-loops not shown

a

b

c

d

e

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