Partial fromvtow v Orders u w implieswalkfromutow AlbertRMeyer - - PowerPoint PPT Presentation

u G+

v AND v G+ w

IMPLIES u G+

w

u R v AND v R w

IMPLIES u R w

• po’s.1

Albert R Meyer March 22, 2013

Mathematics for Computer Science

MIT 6.042J/18.062J

Partial Orders

po’s.2 Albert R Meyer March 22, 2013

Walks in digraph G

walk from u to v and from v to w implies walk from u to w

u v w

po’s.3 Albert R Meyer March 22, 2013

Walks in digraph G

walk from u to v and from v to w, implies walk from u to w: u G+ v AND v G+ w

IMPLIES u G+ w

po’s.4 Albert R Meyer March 22, 2013

Walks in digraph G

transitive relation R: G+ is transitive u R v AND v R w

ES u R w

G+ v AN D v G+ w u G+ w

R v AND v R w

IMPLI

u R w

u R v IMPLIES NOT(v R u)

• po’s.5

Albert R Meyer March 22, 2013

Theorem:

R is a transitive iff R = G+ for some

digraph G

transitivity

po’s.6 Albert R Meyer March 22, 2013

Paths in DAG D

pos length path from u to v implies no path from v to u u D+ v IMPLIES NOT(v D+ u)

po’s.7 Albert R Meyer March 22, 2013

Paths in DAG D

asymmetric relation R:

D+ is asymmetric

y u R v IMPLIES NOT(v R u) strict partial orders

transitive & asymmetric

Albert R Meyer March 22, 2013 po’s.8

R v IMPLIES N OT(v R u)

• po’s.9

Albert R Meyer March 22, 2013

examples:

• ⊂ on sets
• “indirect prerequisite” on

MIT subjects

• less than, <, on real

numbers

strict partial orders

po’s.10 Albert R Meyer March 22, 2013

Theorem:

R is a SPO iff R = D+ for some

DAG D

strict partial orders

po’s.11 Albert R Meyer March 22, 2013

linear orders

Given any two elements,

• ne will be “bigger than”

the other one.

po’s.12 Albert R Meyer March 22, 2013

linear orders

basic example:

< or ≤ on the Reals:

if x ≠ y, then either x < y OR y < x

if x ≠ y, then either x R y OR y R x

• po’s.13

Albert R Meyer March 22, 2013

linear orders

R is linear:

OR no incomparable elements

po’s.14 Albert R Meyer March 22, 2013

The whole partial order is a chain

linear orders

po’s.15 Albert R Meyer March 22, 2013

A topological sort turns a partial order into a linear order

linear orders

…in a way that is consistent with the partial order

po’s.16 Albert R Meyer March 22, 2013

weak partial orders same as a strict partial

• rder R, except that

a R a always holds

examples:

≤ is weak p.o. on R ⊆ is weak p.o. on sets

u R v IMPLIES NOT(v R u) for u ≠ v

• po’s.17

Albert R Meyer March 22, 2013

reflexivity relation R on set A is reflexive iff a R a for all a ∈ ∈A G* is reflexive

po’s.18 Albert R Meyer March 22, 2013

binary relation R is antisymmetric iff it is asymmetric except for a R a case.

antisymmetry

po’s.19 Albert R Meyer March 22, 2013

A/Antisymmetry minor difference: whether aRa is allowed

sometimes

never

po’s.20 Albert R Meyer March 22, 2013

antisymmetric relation R:

D* is antisymmetric for

DAG D

antisymmetry

y y u R v IMPLIES NOT(v R u) for u ≠ v R vIMPLI ES NOT(v R u)

u ≠ v

• po’s.21

Albert R Meyer March 22, 2013

transitive, antisymmetric & reflexive

weak partial orders

po’s.22 Albert R Meyer March 22, 2013

Theorem:

R is a WPO iff R = D* for some

DAG D

weak partial orders

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