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.

1. Redefining RELATION
Putcha V. Narasimham
kenablersys@yahoo.com

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
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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
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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