1. CSC-2259 Discrete Structures
Relations
CSC-2259 Discrete Structures
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2. Relations and Their Properties
A binary relation from set to
is a subset of Cartesian product
Example:
A relation:
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3. A relation on set is a subset of
Example:
A relation on set :
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4. Reflexive relation on set :
Example:
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5. Symmetric relation :
Example:
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6. Antisymmetric relation :
Example:
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7. Transitive relation :
Example:
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8. Combining Relations
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9. Composite relation:
Note:
Example:
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10. Power of relation:
Example:
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11. A relation is transitive if an only if for all
Theorem:
A relation is transitive
if an only if
for all
Proof:
1. If part:
2. Only if part: use induction
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12. Definition of composition:
1. If part:
We will show that if
then is transitive
Assumption:
Definition of power:
Definition of composition:
Therefore, is transitive
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13. We will show that if is transitive then for all
2. Only if part:
We will show that if is transitive
then for all
Proof by induction on
Inductive basis:
It trivially holds
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14. Inductive hypothesis:
Assume that
for all
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15. Inductive step: We will prove Take arbitrary We will show
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16. End of Proof definition of power definition of composition
inductive hypothesis
is transitive
End of Proof
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17. n-ary relations An n-ary relation on sets
is a subset of Cartesian product
Example:
A relation on
All triples of numbers with
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18. (all entries are different)
Relational data model
n-ary relation is represented with table
fields
R: Teaching assignments
Professor
Department
Course-number
Cruz
Zoology
335
412
Farber
Psychology
501
617
Rosen
Comp. Science
518
Mathematics
575
records
primary key
(all entries are different)
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19. keeps all records that satisfy condition
Selection operator:
keeps all records that satisfy condition
Example:
Result of selection operator
Professor
Department
Course-number
Farber
Psychology
501
617
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20. Keeps only the fields of
Projection operator:
Keeps only the fields of
Example:
Professor
Department
Cruz
Zoology
Farber
Psychology
Rosen
Comp. Science
Mathematics
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21. Concatenates the records of and where the last fields of
Join operator:
Concatenates the records of and
where the last fields of
are the same with the first fields of
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22. S: Class schedule Department Course-number Room Time Comp. Science 518
2:00pm
Mathematics
575
N502
3:00pm
611
4:00pm
Psychology
501
A100
617
A110
11:00am
Zoology
335
9:00am
412
8:00am
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23. J2(R,S) Professor Department Course Number Room Time Cruz Zoology 335
9:00am
412
8:00am
Farber
Psychology
501
3:00pm
617
A110
11:00am
Rosen
Comp. Science
518
N521
2:00pm
Mathematics
575
N502
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24. Representing Relations with Matrices
Relation Matrix
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25. Reflexive relation on set :
Diagonal elements must be 1
Example:
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26. Matrix is equal to its transpose:
Symmetric relation :
Matrix is equal to its transpose:
Example:
For all
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27. Antisymmetric relation :
Example:
For all
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28. Union :
Intersection :
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29. Boolean matrix product
Composition :
Boolean matrix product
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30. Boolean matrix product
Power :
Boolean matrix product
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31. Digraphs (Directed Graphs)
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32. there is a path of length from to in
Theorem:
if and only if
there is a path of length
from to in
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33. Connectivity relation:
if and only if
there is some path (of any length)
from to in
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34. Theorem:
Proof:
if then
for some
Repeated node
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35. Closures and Relations
Reflexive closure of :
Smallest size relation that contains
and is reflexive
Easy to find
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36. Smallest size relation that contains and is symmetric
Symmetric closure of :
Smallest size relation that contains
and is symmetric
Easy to find
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37. Transitive closure of :
Smallest size relation that contains
and is transitive
More difficult to find
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38. is the transitive Closure of
Theorem:
is the transitive Closure of
is transitive
Proof:
Part 1:
Part 2:
If and is transitive
Then
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