CS103
HO#24
SlidesFunctions
April 21, 2010
1
Closure
The
closure
of a relation R on a set A with respect to a property P
is the smallest relation containing R that has property P.
R = {( , ), ( , ), .
..( ,)}
A
×
A = {( , ), ( , ), .
..( ,),
(
, ), ( , ), .
..( ,),
(
, ), ( , ), .
..( ,)}
Reflexive closure
: add enough pairs to make the result reflexive
The reflexive closure of R is the relation S = R
∪
{(x, x)  x
∈
A}
Symmetric closure
: add enough pairs to make the result symmetric
The symmetric closure of R is the relation S = R
∪
R
1
= R
∪
{(y, x)  (x, y)
∈
R}
Closure
Transitive closure
: add enough pairs to make the result transitive
S = {(a, b)
∈
A
×
A  there is a path of any length from a to b in R}
Closure
Transitive closure
: add enough pairs to make the result transitive
S = {(a, b)
∈
A
×
A  there is a path of any length from a to b in R}
S = R
1
∪
R
2
∪
R
3
∪
R
4
∪
...
∪
R
n
where n is the number of elements in A
Closure
Transitive closure
: add enough pairs to make the result transitive
R* = {(a, b)
∈
A
×
A  there is a path of any length from a to b in R}
R* = R
1
∪
R
2
∪
R
3
∪
R
4
∪
...
∪
R
n
where n is the number of elements in A
We don't need to go farther than R
n
.
Since there are n elements in A,
a path longer than n would contain a loop, which could be deleted.
n + 1 hops
Since there are only n elements in A, there must be a
duplication, and a shorter path has already been found.
SEA
SFO
DEN
ORD
IDC
JFK
ATL
LA
BOS
SEA
SFO
DEN
ORD
IDC
JFK
ATL
LA
BOS
R
1
∪
R
2
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View Full DocumentCS103
HO#24
SlidesFunctions
April 21, 2010
2
SEA
SFO
DEN
ORD
IDC
JFK
ATL
LA
BOS
R
*
: Transitive closure
KB
In a movie with Kevin Bacon
In a movie with someone who was
in a movie with Kevin Bacon
You
Your friends
Friends of your friends
Modeling Social Networks
Functions
Suppose that after curving,
we use the final numeric averages
to assign letter grades from the set {A, B, C, D, NP}.
We would say that the grade assignment scheme gives us a
mapping
from number grades to letter grades, and that we have a
function
that maps numbers to grades.
95
88
74
92
87
A
B
C
D
NP
Functions
A
function
f is a mapping from a set D to a set T with the property that
for each element d in D, the function f maps d to a single element of T,
denoted f(d).
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 Fall '09
 Kevin Bacon, C D NP

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