Fuzzy Implications

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FUZZY IMPLICATIONS
Summer Project
MASTER OF SCIENCE
IN
APPLIED MATHEMATICS
By
Himani Jain
Enrollment No: A4451014015
Under the Supervision of:
Dr. Rashmi Singh

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In real world, there exists much fuzzy knowledge;
Knowledge that is vague, imprecise, uncertain, ambiguous,
iexact, or probabilistic in nature. Fuzzy sets have been
introduced by Lofti A. Zadeh (1965) as an extension of the
classical notion of set
Human thinking and reasoning frequently involve fuzzy
information, originating from inherently inexact human concepts.
Humans, can give satisfactory answers, which are probably true.
However, our systems are unable to answer many questions.
The reason is, most systems are designed based upon classical
set theory and two-valued logic which is unable to cope with
unreliable and incomplete information and give expert opinions.
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INTRODUCTION

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We want, our systems should also be able to cope with
unreliable and incomplete information and give expert
opinions. Fuzzy sets have been able to provide solutions to
many real world problems.
Fuzzy set theory is an extension of classical set theory where
elements have degrees of membership.
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4. • The use of linguistic variables and fuzzy propositions in the fuzzy contexts
can be an interesting tool to extract a more complete information from them.
• Uncertainty -- Arising from imprecision, vagueness, non-specificity,
inconsistency
○ Traditional view:
-- Uncertainty is undesirable and should be avoided.
○ Modern view:
-- Uncertainty is not only an unavoidable plague, but it has a
great utility.
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Crisp Sets
○ Methods for describing sets
1. Enumeration
2. Description
3. Characteristic function
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Fuzzy Sets: Basic Types
○ Fuzzy sets
－Sets with vague boundaries
－Membership of x in A is a matter of degree to
which x is in A
○ Utilization of fuzzy sets
(1) Representation of uncertainty
(2) Representation of conceptual entities
e.g., expensive, close, greater, sunny, tall
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8. ○ Fuzzy Sets Crisp Sets
membership characteristic
function function
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◎ Fuzzy Set Operations
-- Standard complement︰
-- Equilibrium points︰
-- Standard intersection︰
-- Standard union:
-- Difference
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CLASSICAL IMPLICTIONS
Classical implications are of two types: conditional and biconditional.
Conditional :
The conditional p  q, which we read "if p, then q" or "p implies q," is
defined by the following truth table.
The arrow “" is the conditional operator, and in p  q the statement p
is called the antecedent, or hypothesis, and q is called
the consequent, or conclusion.
p q p  q
T T T
T F F
F T T
F F T

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Biconditional :
The biconditional p q, which we read "p if and only if q" or "p is
equivalent to q," is defined by the following truth table.
↔
p q p q
T T T
T F F
F T F
F F T
↔
The arrow " " is the biconditional operator.
Note that, from the truth table, we see that, for p q to be true, both p and q
must have the same truth values; otherwise it is false.
↔
↔

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FUZZY IMPLICATION
Definition 1 A binary operator �� on [0, 1] is called to be an
im plicatio n functio n (or an im plicatio n) if it satisfies :
i) �� ሺ�, �ሺ≥ ��(�, �) �ℎ𝑒� � ≤ �, for all � ∈ [0, 1];
ii) ��(�, �) ≤ �� (�, �) �ℎ𝑒� � ≤ �, for all � ∈ [0, 1];
iii) �� (0, 0) = �� (1, 1) = �� (0, 1) = 1 and �� (1, 0) = 0 etc.

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Definition 2 We say that the following properties are satisfied by a
fuzzy implication ��:
1) The le ft ne utra lity pro pe rty (NP), if
�� (1, �) = � , � ∈ [0,1]
2) The o rde ring pro pe rty (OP), if
� ≤ � ՞ ��(�, �) = 1, �, � ∈ [0, 1]
3) The ide ntity pro pe rty (IP), if
�� (�, �) = 1 , �, � [0, 1]

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4) The e xcha ng e pro pe rty (EP), if
�� (�, �� (�, �)) = �� (�, �� (�, �)) , �, �, � [0, 1]
5) The law o f co ntrapo sitio n with respect to a fuzzy negation
�, (�(�)), if
�� ሺ�, �ሺ= ��(�(�), �(�)) , �, �, � [0, 1]
Some basic fuzzy implications are in next slide…

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LATTICE ON FUZZY IMPLICATION
Definition 1 A partially o rde re d re latio n is a relation ≤ which is
defined on any fuzzy implication and satisfies following
properties :
i) Reflexive.
ii) Antisymmetric.
iii) Transitive.
And (��, ≤) is called a po se t.

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Definition 2 A lattice o f a fuz z y im plicatio n is a partial ordered implication
(��, ≤) in which every pair of implication �, � ∈ �� has a greatest lower
bound (meet) and least upper bound (join)
∧ {�, �} = 𝑚𝑒𝑒�{�, �} = 𝐺𝑙� {�, �} = {� (�, �) ∩ � (�, �)} = 𝑚�� {�, �}
∨ {�, �} = 𝑗𝑜��{�, �} = �𝑢� {�, �} = {� (�, �) ∪ � (�, �)} = 𝑚𝑎� {�, �}
A Fuzzy implication lattice will always be nonempty; but it can consist of
only one element - the smallest element coincides with the largest one.

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SOME PROPERTIES OF FUZZY IMPLICATION
LATTICE
Following are some properties of the two binary
operations of max and min on a fuzzy implication lattice
(��, ≤) , for L, M, N ∈ �� we have
I. Idempotent Law
𝑚�� (� (�, �), � (�, �)) = � (�, �)
𝑚𝑎� (� (�, �), � (�, �)) = � (�, �)

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II. Commutative Law
𝑚�� (� (�, �), � (�, �)) = 𝑚�� (� (�, �), � (�, �))
𝑚𝑎� (� (�, �), � (�, �)) = 𝑚𝑎� (� (�, �), � (�, �))
III. Associative Law
𝑚��(𝑚��(�(�, �), �(�, �)), �(�, �))
= 𝑚��(�(�, �), 𝑚��(�(�, �), �(�, �)))
𝑚𝑎�(𝑚𝑎�(�(�, �), �(�, �)), �(�, �))
= 𝑚𝑎�(�(�, �), 𝑚𝑎�(�(�, �), �(�, �)))

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IV. Absorption Law
𝑚�� (𝑚𝑎� (�(�, �), �(�, �)), �(�, �)) = �(�, �)
𝑚𝑎� (𝑚�� (�(�, �), �(�, �)), �(�, �)) = �(�, �)
In the following slides we show an example of fuzzy
implication lattice.

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EXAMPLE : Let (��, ≤) be a fuzzy implication. Their
greater lower bound is max and lowest upper bound is
min. Let ���ሺ�, �ሺ, � 𝑅�ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ∈ ��. Now we will
show that these satisfy all the above four properties
where �, � ∈ [0, 1] and
NAME FORMULA
Lukasiewicz ILK ሺx, yሺ= ( 1, 1 − x + y)
Rescher
IRS ሺx, yሺ= ሺ
1, if x ≤ y
0, if x > �
Weber IWB ሺx, yሺ= ሺ
1, if x < 1
y, if x = 1

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SOLUTION Let x = 0.7 and y = 0.6 ,then
���ሺ0.7, 0.6ሺ= minሺ1, 1 − 0.7 + 0.6ሺ= minሺ1, 0.9ሺ= 0.9
� 𝑅�ሺ0.7, 0.6ሺ= 0 (𝑎� � > �)
� 𝑊� ሺ0.7, 0.6ሺ= 1 (𝑎� 0.7 < 1)

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Now,
1) Idempotent Law :
• 𝑚�� (���ሺ�, �ሺ, ���(�, �)) = 𝑚��(0.9, 0.9) = 0.9 =
���(�, �)
• 𝑚𝑎� (���ሺ�, �ሺ, ���(�, �)) = 𝑚𝑎�(0.9, 0.9) = 0.9 =
���(�, �)

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2) Commutative Law :
• L. H. S = min൫�𝐿�ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ൯= minሺ0.9, 1ሺ
= 0.9
= minሺ1, 0.9ሺ= min൫� 𝑊�ሺ�, �ሺ, �𝐿�ሺ�, �ሺ൯= �. �. �
• 𝐿. �. � = max൫�𝐿�ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ൯= maxሺ0.9, 1ሺ
= 1
= maxሺ1, 0.9ሺ= 𝑚𝑎�൫� 𝑊�ሺ�, �ሺ, �𝐿�ሺ�, �ሺ൯= �. �. �

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P a g e | 13
3) Associative Law :
• 𝐿. �. � = min ሺmin൫ILK ሺx, yሺ, IRS ሺx, yሺ൯, IWB ሺx, yሺሺ
= minሺminሺ0.9, 0ሺሺ, 1)
= minሺ0, 1ሺ
= 0
�. �. � = min⁡(ILK ሺx, yሺ, min⁡(IRS ሺx, yሺ, IWB ሺx, yሺ))
= minሺ0.9, minሺ0,1ሺሺ
= minሺ0.9, 0ሺ
0

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• L. H. S = max⁡(max൫ILK ሺx, yሺ, IRS ሺx, yሺ൯, IWB ሺx, yሺ)
= maxሺmax⁡(0.9, 0ሺ), 1)
= maxሺ0.9, 1ሺ
= 1
�. �. � = max൫�𝐿�ሺ�, �ሺ, max൫���ሺ�, �ሺ, � 𝑊�ሺ�, �ሺ൯൯
= maxሺ0.9, maxሺ0,1ሺሺ
= maxሺ0.9,1ሺ
= 1

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P a g e | 15
4)Absorption Law :
• L. H. S = min ሺmax൫ILK ሺx, yሺ, IRS ሺx, yሺ൯, IWB ሺx, yሺሺ
= minሺmaxሺ0.9,0ሺ, 1ሺ= minሺ0.9,1ሺ= 0.9
= ILK ሺx, yሺ= �. �. �
• 𝐿. �. � = max ሺmin൫� 𝐿� ሺ�, �ሺ, ��� ሺ�, �ሺ൯, � 𝑊� ሺ�, �ሺሺ
= maxሺminሺ0.9,0ሺ, 1ሺ= maxሺ0,1ሺ= 1
= � 𝑊� ሺ�, �ሺ= �. �. �

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