Relation in set theory of math is flawed. It remains undetected and uncorrected even now. This 11 slide PPT point outs the errors with examples, corrects them, giving a better definition. It may be too elementary but how can math live with errors in fundamentals?

1. Relation Definition—Flaws,
Correction, and REDEFINITION
Putcha V. Narasimham
putchavn@yahoo.com

2. ..asking questions is more valuable …..
In mathematics*, the art of asking questions is more
valuable than solving problems
Georg Cantor
* Well, not only in mathematics?
December 23, 2020Copyright (C) 2004-2020 Putcha V. Narasimaham
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3. Relationship
 A relationship is a connection among things
The UML User Guide, Too casual
 There are many definitions but they seem to be flawed, See
 https://www.slideshare.net/putchavn/relation-need-for-radical-redefinition
 https://www.slideshare.net/putchavn/relation-need-for-radical-redefinition-pdf
 See the definition from Wolfram Mathworld and
 Criticism and redefinition of “relation” by PVN
Copyright (C) 2004-2020 Putcha V. Narasimaham
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December 23, 2020

4. Relation Vs Relationship
While relation and relationship
refer to the connection between
two things, relation shades more
toward the way things are
connected,
while relationship refers to the
connection itself. The difference
is not spacious.
 Relationship is social and
relation is technical !
 Relationship is bonding and
relation is a connection !!
 There are more and conflicting
explanations
 In technical texts relation is
widely use (we do the same)
December 23, 2020Copyright (C) 2004-2020 Putcha V. Narasimaham
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5. Definition of Relation--Two Flaws per PVN
A relation is any subset of a Cartesian product.
For instance, a subset of A X B , called a "binary
relation from A to B," is a collection of ordered
pairs (a, b) with first components from A and
second components from B, and, in particular, a
subset of AXA is called a "relation on A." For a
binary relation R, one often writes aRb to mean
that (a, b) is in RxR.
http://mathworld.wolfram.com/Relation.html
December 23, 2020Copyright (C) 2004-2020 Putcha V. Narasimaham
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This is any subset of all
possible ordered pairs
but still not a definition
of what a relation is
This is a procedure of
forming a subset of all
possible ordered pairs
without any criteria for
selecting a’s and b’s for
a specific relation

6. A set of the related is NOT a Relation
Any ‘collection of
ordered pairs (a,b)’
 is only a ‘set of related
elements’
 but NOT relation
Copyright (C) 2004-2020 Putcha V. Narasimaham
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December 23, 2020

7. Relation without Basis is NOT a relation
Any collection
of ordered pairs
cannot be a
relation
 If there is no basis to form ordered
pairs, the relation becomes indefinite,
arbitrary & meaningless
 The basis, even if it exists, cannot be
derived from a single set of ordered
pairs
 The basis must be stated explicitly
Copyright (C) 2004-2020 Putcha V. Narasimaham
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December 23, 2020

8. Proposed Correction by PVN
A relation must have
a meaningful name
with criteria for
forming the ordered
pairs of the relation
 The relations “Brother & Sister” and
“Husband & Wife” are formed from
the same sets, Men and Women
 Pairs alone do not reveal relation
 Criteria of pairing are more crucial
for relation than pairs themselves
Copyright (C) 2004-2020 Putcha V. Narasimaham
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December 23, 2020

9. Ill-defined relation, its flaws, and corrections
The ill-defined relation
 Needs predefined sets
with existing members
to choose the
members of ordered
pairs from
 Does not exist if there
is no set of ordered
pairs
 Notion of relation arises by identifying sets to be
related and how to relate them
 A relation is defined by a meaningful name and
criteria of forming ordered pairs from identified sets
 Relation exists without actually forming ordered pairs
 Relation applies to evolving sets also
Copyright (C) 2004-2020 Putcha V. Narasimaham
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December 23, 2020

10. Non-existing members can be related
 Children of the same parents
are related as “siblings”
 This applies to the existing &
to-be-born children
 The ill-defined relation
excludes them
The need to actually form and
enumerate all ordered pairs of a
relation, excludes the potentially-
relatable members when they
become valid members of
identified sets in the future
Copyright (C) 2004-2020 Putcha V. Narasimaham
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December 23, 2020

11. Relation REDEFINED
 A relation between a set A and Set B is a set
 with a meaningful name, and
 principles & criteria for forming ordered pairs from
A to B
• See the PDF and PPT on slideshare for examples and reasons
December 23, 2020Copyright (C) 2004-2020 Putcha V. Narasimaham
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