CSE390D Notes for Wednesday, 11/20/24
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Chapter 9--relations
motivation...talk about history of databases...fancy table manipulation
how to track person + preferred email? table of 2 columns
how to add an address? extra columns for address
how to add multiple addresses?
extra columns--bad--not flexible--pedit story (McCarthy/fascist)
2nd table: person-id + address
how to keep track of students and classes and enrollments?
table for student info (with key)
table for course info (with key)
table for enrollment (two keys, possibily other data like S/NS)
That third table is a relation.
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Definition: A relation from A to B is a subset of AxB
not the same as a function (just think about the many-to-many mapping of
students enrolled in courses)
Definition: A relation on A is a relation from A to A
example: facebook friends
Relations have properties:
reflexive: (a, a) in R for all a in A
example: has talked to
symmetric: (b, a) in R whenever (a, b) in R for all a,b in A
example: facebook friends
antisymmetric:
example: less-than
might think to say (b, a) not in R whenever (a, b) in R for all a,b in A
not quite...what about reflexive (e.g., <=)?
instead: whenever (a, b) in R and (b, a) in R for a, b in A, a=b
transitive: whenever (a, b) in R, (b, c) in R, (a, c) in R for a,b,c in A,
example: <, <=, >, >=
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Combining relations
Because relations from A to B are subsets of A x B, two relations
from A to B can be combined in any way two sets can be combined.
Example
Let A and B be the set of all students and the set of all courses
at a school, respectively. Suppose that R1 consists of all
ordered pairs (a, b), where a is a student who has taken course b,
and R2 consists of all ordered pairs (a, b), where a is a student
who requires course b to graduate. What are the relations:
R1 U R2 courses the student has taken and has to take
R1 intersect R2 required courses the student has taken
R1 - R2 courses the student took that weren't required
R2 - R1 required courses the student hasn't yet taken
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Definition:
Let R be a relation from the set A to a set B and S a relation
from B to a set C. The composite of R and S is the relation
consisting of ordered pairs (a, c) where a in A, c in C, and for
which there exists an element b in B such that (a, b) in R and (b,
c) in S. We denote the composite of R and S by S o R.
Example:
(a, b) in Parent: b is a parent of a
(a, b) in Sister: b is a sister of a
What is Parent o Sister? parent of a sister
What is Sister o Parent? aunt
S o R = {(a, c) | exists b such that (a,b) in R and (b,c) in S}
Example:
Using the relations: Parent, Child, Brother, Sister, Sibling,
Father, Mother express
Uncle: b is an uncle of a Brother o Parent
Cousin: b is a cousin of a Child o Sibling o Parent
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Definition
Let R be a relation on the set A. The powers R^n, n = 1, 2, 3,
..., are defined recursively by:
R1 = R and R^(n+1) = R^n o R
Theorem:
The relation R on a set A is transitive if and only if R^n is a
subset of R for n = 1, 2, 3, ...
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Stuart Reges
Last modified: Thu Nov 21 22:30:04 PST 2024