Relation.....too fundamental to be ill-defined but is it well defined? Is it useful? Not quite according to me. I have my questions and reasons to redefine relation radically. Here is the proposal.
2. What is a relation?
• Well known but difficult to define
• One ends up using association or connection
• They are just equivalent to relation
• But NOT definitions!
• Let’s look up online dictionary
• http://dictionary.reference.com/browse/relation?s=t
22JUL13Need to Redefine RELATION
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Relation
What is it?
3. re·la·tion
• noun
• 1. an existing connection; a significant association
between or among things
• 2. connections in which persons or things are brought
together
• Well these are NOT precise and rigorous enough
• But one can get an idea of what a relation is
• Let’s see Math definition
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How is rain
related to clouds?
4. Relation—Math definition
A relation is any subset
of a Cartesian product
Consider two sets A & B
Then, relation is a subset
of CP of A&B
22JUL13Need to Redefine RELATION
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Relation:
Subset of CP
of A & B
Set A
Men
Set B
Women
5. Cartesian Product Definition
• Cartesian Product of two sets A & B
• Is a set of all ordered pairs <a,b>,
• Where
• ‘a’ is an element of A &
• ‘b’ is an element of B
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Set A
Men
Set B
Women
6. Cartesian Product Explanation
• Every member of A
• is paired with
• every member of B
• To generate CP
• CP is the full set of all
possible pxq ordered
pairs
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Set A
Men
Set B
Women
M1
Mp
W1
W2
Wq
7. Relation: A subset of Cartesian Product
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Relation
Set A
Men
Set B
Women
M1, M2, M3, M4,
M5, M6, M7 … M20
W1, W2, W3, W4,
W5, W6, W7… W30
Subset
M2, W4
M4, W7
M5, W10
M9, W1
M13, W27
8. Ordered Pairs OK, but is it a relation?
1. We only see WHO are related
2. But don’t know
WHAT the relation is?
WHAT brings them into a relation?
And, HOW are they related?
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Subset
M2, W4
M4, W7
M5, W10
M9, W1
M13, W27
9. Is it a relation at all?
• It is just a set of ordered pairs
• WHAT is the basis to form pair?
• If any set is a relation, how is one
relation different from another, except
for the exact pairs?
• Is it not possible that the same pairs
are also related differently?
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Subset
M2, W4
M4, W7
M5, W10
M9, W1
M13, W27
10. Ordered Pairs & Relations
• Related Ordered Pairs
• Father and daughter,
• Brother and sister
• Husband and wife
• Mixed pairs for games
• Pairing alone is not relation
• Basis for pairing is relation
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Relation is
the basis for
pairing
NOT just
pairs
11. Arbitrary pairs remain unrelated
• One can form them but
what purpose do they
serve?
• What meaning or
significance do they
have?
• In businesses, societies and nature
there are always principles and
criteria guiding the formation of
classes (categories) and relations
• They are important
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12. Classes and Relations need basis
• This is the
motivation for
this proposal
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Classes and relations need a set
of principles and criteria for the
their formation as in OOAD, UML
They have identities and
properties for use and application
13. Relation is as fundamental as set
Is a member of is a relation
between an object and the set to
which it belongs
The concept of relation and set
perhaps emerge interdependently
and simultaneously
This may have been
sorted out in some
fundamental theories
of mathematics and or
philosophy
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22JUL13Need to Redefine RELATION
14. Relation Redefined for clarity and use
• The mathematical definition of relation
appears unmathematical and leaves many
questions unanswered
• It is enumerative & unusable
• Runs into anomalies and paradoxes
• So, here is a proposed definition of relation
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15. Proposed Definition of relation
A relation between two sets A and
B is a named set of principles and
criteria, according to which
ordered pairs can be formed from
A and B.
Criteria for forming
ordered pairs are
more important
than the pairs
This resolves many
questions
See PDF--Relation:
Need for Radical
Redefinition
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22JUL13Need to Redefine RELATION
16. How does this definition help?
• Now, we know what a relation is and how to
use it without forming ordered pairs first
• We know how relation applies for the FUTURE
members of sets involved
• We can use it to define, create, VERIFY and
apply relations correctly and infer from them
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17. Let’s THINK and Discuss
• A definition should be clear, consistent
and self-sufficient for application and
use.
• I believe the proposed definition is
• I welcome reviews, examples,
counter examples and feedback
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START
kenablersys@yahoo.com