Intro Consider the following example telephone line bus route a b c d Is R defined above on the set Aa b c d transitive If not is there a possibly indirect link between each of the cities ID: 467243
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Slide1
8.4 Closures of Relations
Slide2
Intro
Consider the following example (telephone line, bus route,…)
a b c
dIs R, defined above on the set A={a, b, c, d}, transitive?If not, is there a (possibly indirect) link between each of the cities?To answer, we want to find the Transitive ClosureSlide3
Closures, in general
Def: Let
R be a relation on a set A that may or may not have some property P. (Ex: Reflexive,…) If there is a relation S with property P containing R such that S is a subset of every relation with property P containing R, then S is called the
closure of R with respect to P. Note: the closure may or may not existSlide4
Reflexive Closures- Idea, Example
Reflexive Closure of R—the smallest reflexive relation that contains R
Consider R={(1,2),(2,3),(3,2)} on A={1,2,3}
1 2 3Using both ordered pairs and digraphs, find the reflexive closure.Slide5
Reflexive Closures
Reflexive Closure of R—the smallest reflexive relation that contains R
Reflexive Closure = R
Where ={(a,a)| a A} is the diagonal relation on A.Slide6
More examples
Find the reflexive closures for:
R={(
a,b)|a<b} on the integers ZR={(a,b)|a ≠ b} on ZSlide7
Symmetric Example
Find the symmetric closure of
R={(1,1), (1,2),(2,2),(2,3),(3,1),(3,2)} on A={1,2,3}
2 3Slide8
Symmetric Closures
Symmetric Closure of R = R
R-1Where R-1= {(b,a) | (a,b) R}
Example:R={(a,b)|a>b} on the integers Z
Symmetric closure: Slide9
Transitive Theory- example
1 2
4 3
Add all (a,c) such that (a,b), (b,c) R.
Keep going. (Why?)Slide10
Transitive Closure Theory, and Def of Path
Def: A
path from a to b in a directed graph G is a sequence of edges (x0,x1), (x1,x2)… (xn-1,
xn) in G where x0=a and x
n=b. It is denoted x1, x2,…x
n and has length n. When a=b, the path is called a circuit or cycle.
Slide11
Find Transitive Closure- see worksheet
Do Worksheet
1 2 4 3 Find the transitive closure
Find circuits and paths of length 2, 3, 4 Slide12
Example- in matrices
Using the idea that R
n+1
= Rn°R and MS°R = MR
MS , Find the matrices for R R
2 R 3 R 4
The find paths of length 2, 3, 4Slide13
Example
= Slide14
Next step
In order to come up with a theory for the transitive closure, we will first study paths….Slide15
Theorem 1
Theorem 1: Let R be a relation on a set A.
There is a path of length n from a to b
iff (a,b)Rn Proof method?Slide16
Proof of Thm
. 1
By induction:
N=1: true by definition (path from a to b of l=1 iff (a,b) R).
Induction step: Assume: There exists a path of length __ from ___iff ______
Show: There exists a path of length __ from ___iff ______Assuming the IH (Inductive Hypothesis),
There is a path of length __ from ___ Iff There exists an element c with a path from a to c in R and a path of length n from c to b
in ___
Iff
There exists an element c with (
a,c
) ___ and (
c,b
) ___
Iff
(
a,b
) ____ = _______Slide17
Def 2: Connectivity relation
Def. 2: Let R be a relation on set A.
The
connectivity relation R* consists of the pairs (a,b) such that there is a path between a and b in R.R* = Slide18
Examples
R
={(
a,b)| a has met b} 6 degrees Erdos numberR* include (you,__)R={(
a,b)| it is possible to travel from stop a to b directly} on set A of all subway stopsR*=R={(a,b
)|state a and b have a common border” on the set A of states. R*=Slide19
Thm
. 2: Transitive closure is the connectivity relation
Theorem 2: The transitive closure of a relation R equals the
connectivity relation R* = Elements of the Proof:Note that
R R*To show R* is the transitive closure of R, show:1) R* is ________
2) Whenever S is a transitive relation that contains R, then R* ______Slide20
Proof of Thm
2
Assume (
a,b) R* and (b,c) R*So (a,b
) ___ and (b,c) ___
By Thm. 1, there exists paths…2 paths:
In conclusion ________Slide21
Thm 2 proof…
2)
Suppose S is a transitive relation containing R
It can be shown by induction that Sn is transitive.By a previous theorem in sec. 8.1, S n
___ S.Since S* = S k and S
k __ S , the S* ___ S.Since R ___S, the R* ____ S*.
Therefore R* ___ S* ___ S.