Relation--Need for Radical Redefinition PPT

Redefining RELATION

Putcha V. [email protected]

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?

re·la·tion

• noun

• 1. an existing connection; a significant associationbetween 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?

http://dictionary.reference.com/browse/which

Relation—Math definition

A relation is any subsetof 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

MenSet B

Women

http://mathworld.wolfram.com/Subset.html

http://mathworld.wolfram.com/CartesianProduct.html

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

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

MenSet B

Women

M1

Mp

W1

W2

Wq

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

SubsetM2, W4

M4, W7

M5, W10

M9, W1

M13, W27

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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SubsetM2, W4

M4, W7

M5, W10

M9, W1

M13, W27

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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SubsetM2, W4

M4, W7

M5, W10

M9, W1

M13, W27

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

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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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

Relation is as fundamental as set

Is a member of is a relationbetween 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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Relation Redefined for clarity and use

• The mathematical definition of relationappears 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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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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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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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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[email protected]

Relation--Need for Radical Redefinition PPT

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Transcript of Relation--Need for Radical Redefinition PPT