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#121
 
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Re: Foreign keys -
01-16-2008
, 07:42 AM
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On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. |
A --> BC IFF A --> B /AND/ A --> C;
if they weren't, then it could only be said that
A --> BC IFF A --> B /OR/ A --> C.
Similarly, the relation schema,
R {A, B, C} where A --> BC,
can represent exactly the same information as the pair of relation schemata,
S {A, B} where A --> B, T {A, C} where A --> C,
if and only if the cyclical inclusion dependency,
S[A] = T[A] (mutual foreign keys)
also holds.
Quote:
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-- -kira |
#122
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#123
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#124
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#125
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#126
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#127
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#128
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#129
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"Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011602453916807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-16 01:11:22 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: "Kira Yamato" <kirakun (AT) earthlink (DOT) net> wrote in message news:2008011600092016807-kirakun (AT) earthlinknet (DOT) .. On 2008-01-15 22:05:24 -0500, "Brian Selzer" <brian (AT) selzer-software (DOT) com said: In relational algebra, say I have a relation Person(Name, Age). Then there is functional dependency Name --> Age. Now, the domain for Age can be any non-negative integer. However, the extensional relation will not have a tuple for every possible value of Age. [...] Are you saying that Name --> Age is not a functional dependency because it is not surjective? No. What I'm saying is that not only is it the case that for every Name in the extension there is one and only one Age, but also that whenever there is an Age in the extension, there must also be a Name. This is due to the fact that Name and Age appear in the same relation schema. So in that sense Name --> Age /is/ surjective. I keep wanting to mention active domains, and for a database with a single relation, it would be valid to say that Name --> Age describes a surjection between the active domain for Name and the active domain for Age, but for databases with multiple relations, an active domain is the set of all values from a domain that are in any relation in the database. Ok. I'm beginning to see your point. So, in the case of a single relation, the surjectivity holds trivially from the definition of functional dependency. However, in the case of foreign key where 2 relations are involved, the foreign key does not necessarily maps to every possible primary key in the other relation. So, it is possible to not have surjectivity between attributes across 2 relations. So, I do have one more question. Why do we want to require functional dependencies to be surjective? What desirable property of relational algebra do we lose if we introduct functional dependencies that are not surjective? Thanks. Since functional dependencies are surjective, it can be said that A --> BC IFF A --> B /AND/ A --> C; |
Database System," defines as Join-dependency.
Quote:
|
if they weren't, then it could only be said that A --> BC IFF A --> B /OR/ A --> C. |
if A --> B and A --> C, then A --> BC.
However, the converse statement is not necessarily true.
Quote:
|
Similarly, the relation schema, R {A, B, C} where A --> BC, can represent exactly the same information as the pair of relation schemata, S {A, B} where A --> B, T {A, C} where A --> C, if and only if the cyclical inclusion dependency, S[A] = T[A] (mutual foreign keys) also holds. |
of representation through joins.
--
-kira
#130
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On 2008-01-14 21:18:57 -0500, "Evan Keel" <evank... (AT) sbcglobal (DOT) net> said: Always a physical issue. Never a theory issue.Agree? Foreign keys are functional dependencies across two relations. More specifically, let * * * * R1(K1, A1, B1) be a relation with attribute sets K1, A1 and B1 where K1 is R1's primary key and B1 is a foreign key to the relation * * * * R2(K2, A2) where K2 is R2's primary key and A2 is the set of its remaining attributes.. Then the foreign key B1 represents the functional dependency * * * * B1 --> A2, which is the functional dependency across two relation I mentioned in the first sentence. Furthermore, through transitivity by the functional dependency K1 -- B1, the foreign key also represents the inter-relational functional dependency * * * * K1 --> A2. Am I correct to say this? |
the same for functional and inclusion dependencies is correct, but
unfortunately this is not true in general. For example, the
augmentation rule does not always hold for inclusion dependencies. In
fact, axiomatizing the combination of functional and inclusion
dependencies is a notoriously difficult problem and in the past there
has been intensive research on that subject.
-- Jan Hidders
Quote:
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-- -kira |
 |
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