Introduction to Fuzzy logic

Introduction: Fuzzy Logic
Adri Jovin J J, M.Tech., Ph.D.
UITE221- SOFT COMPUTING

Soft Computing
• Introduced by Lotfi A. Zadeh, University of California, Berkley
• Collection of computational methods
• Includes Fuzzy Systems, Neural Networks and Evolutionary Algorithms
• Deployment of soft computing for the solution of machine learning problems has led to high Machine Intelligence
Quotient
UITE221 SOFT COMPUTING 2
Image Credit: Electrical Engineering and Computer Sciences, UC, Berkeley
“Soft computing differs from hard computing (conventional computing) in its tolerance to
imprecision, uncertainty and partial truth”
-Lotfi A. Zadeh

Soft Computing (Contd…)
Fuzzy Systems
Neural
Networks
Evolutionary
Algorithms
Fuzzy-evolutionary hybrids Neuro-fuzzy hybrids
Neuro-evolutionary hybrids
Neuro-fuzzy-evolutionary hybrids

Fuzzy Logic
“As the complexity of a system increases, it becomes more difficult and
eventually impossible to make a precise statement about its behavior, eventually arriving
at a point of complexity where the fuzzy logic method born in humans is the only way to
get at the problem.”
-Lotfi A. Zadeh
Image Credit: Electrical Engineering and Computer Sciences, UC, Berkeley

Fuzzy Logic (Contd.)
Introduced in they year 1965
Japanese have utilized the full potential of fuzzy sets by commercializing the technology
Fuzziness means “vagueness”
Mathematical tool to handle uncertainty arising due to vagueness
Understanding human speech, handwriting recognition

Fuzzy Logic (Contd…)
Fuzz Logic System
Imprecise and
vague data Decisions
0.5
1.0 Tall
150 180 210
Membership
Height (cm)
0.5
1.0 Tall
150 180 210
Membership
Height (cm)
Short Medium

• Describe tall or short or medium height…
• “short” and “tall” are linguistic variables
• Set membership helps appropriately to distinguish linguistic variables
• Various degree of membership on a real continuous interval [0,1]
• Fuzzy sets accommodate the degrees of membership
This Photo by Unknown Author is
licensed under CC BY-SA-NC

• A fuzzy set 𝐴 contains an object 𝑥 to degree 𝑎(𝑥)
• 𝑎 𝑥 = 𝐷𝑒𝑔𝑟𝑒𝑒(𝑥 ∈ 𝐴) and the map 𝑎: 𝑋 → {𝑀𝑒𝑚𝑏𝑒𝑟𝑠ℎ𝑖𝑝 𝐷𝑒𝑔𝑟𝑒𝑒𝑠} is called a set function or a membership function
• Fuzzy set 𝐴 can be expressed as A = 𝑥, 𝑎 𝑥 , 𝑥 ∈ 𝑋 which defines the possibility distribution
• Fuzzy sets form the building blocks for fuzzy IF-THEN rules which is of general form “IF X is A THEN Y is B”
• Fuzzy systems refer to the systems governed by fuzzy IF-THEN rules
• IF part of the implication is called antecedent and THEN part is called precedent
• Possess partial matching capability

• Rule based system constructed from the collection of linguistic rules on one hand
• Non-linear mappings of inputs (stimuli) to outputs (response) on the other hand
• Inputs and outputs can be numbers or vectors of numbers
• Rule-based systems can be any system with arbitrary accuracy, i.e., they work as universal approximators
• Smart rules give smart system
• Number of rules increases exponentially with the dimension of the input space
• Rule explosion is called the curse of dimensionality

Classical sets (Crisp sets)
• Set is a collection of objects sharing certain characteristics
• No partial membership exist in crisp sets
• Crisp set is defines by its characteristic function

Universe of discourse
• Also known as universal set (U)
• Contains all possible elements having the same characteristics, from which sets can be formed
• Crisp set A in universe U
• An object 𝑥 is a member of given set 𝐴 (𝑥 ∈ 𝐴) ; 𝑥 belongs to 𝐴
• An object x is not a member of given set A (𝑥 ∉ 𝐴); x does not belong to A
U
A

Defining a set
• List of all the members of a set may be given
𝐴 = 2,4,6,8,10
• The properties of the set of elements may be specified
𝐴 = {𝑥|𝑥𝑖𝑠 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟 ≤ 10}
• The formula for the definition of a set may be mentioned
𝐴 = 𝑥𝑖 =
𝑥𝑖 + 1
5
, 𝑖 = 1 𝑡𝑜 10, 𝑤ℎ𝑒𝑟𝑒 𝑥𝑖 = 1

Defining a set (Contd…)
• Basis of the results of a logical operation
𝐴 = 𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑏𝑒𝑙𝑜𝑛𝑔𝑖𝑛𝑔 𝑡𝑜 𝑃 𝐴𝑁𝐷 𝑄
• There exist a membership function, usually denoted by 𝜇
𝜇𝐴 𝑥 =
1 𝑖𝑓 𝑥𝜖 𝐴
0 𝑖𝑓 𝑥 ∉ 𝐴
• Empty set or null set is usually denoted by 𝜙, which indicates the occurrence of an impossible event
• Set containing the possible subsets of a given set A is called a power set
𝑃 𝐴 = {𝑥|𝑥 ⊆ 𝐴}

Operations on Classical Sets: Union
𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}
A B

Operations on Classical Sets: Intersection
𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}
A B

Operations on Classical Sets: Complement
𝐴 = {𝑥|𝑥 ∉ 𝐴 , 𝑥 ∈ 𝑈}
A

Operations on Classical Sets: Difference
A B

Properties of Classical Sets
Commutativity
𝐴 ∪ 𝐵 = 𝐵 ∪ 𝐴; A ∩ 𝐵 = 𝐵 ∩ 𝐴
Associativity
𝐴 ∪ 𝐵 ∪ 𝐶 = 𝐴 ∪ 𝐵 ∪ 𝐶; 𝐴 ∩ 𝐵 ∩ 𝐶 = (𝐴 ∩ 𝐵) ∩ 𝐶
Distributivity
𝐴 ∪ 𝐵 ∩ 𝐶 = 𝐴 ∪ 𝐵 ∩ 𝐴 ∪ 𝐶
𝐴 ∩ 𝐵 ∪ 𝐶 = (𝐴 ∩ 𝐵) ∪ (𝐴 ∩ 𝐶)

Properties of Classical Sets (Contd…)
Idempotency
𝐴 ∪ 𝐴 = 𝐴; 𝐴 ∩ 𝐴 = 𝐴
Transitivity
𝐼𝑓 𝐴 ⊆ 𝐵 ⊆ 𝐶, 𝑡ℎ𝑒𝑛 𝐴 ⊆ 𝐶
Identity
𝐴 ∪ 𝜙 = 𝐴; 𝐴 ∩ 𝜙 = 𝐴
𝐴 ∪ 𝑋 = 𝑋; 𝐴 ∩ 𝑋 = 𝑋

Properties of Classical Sets (Contd…)
Involution
𝐴 = 𝐴
Law of excluded middle
𝐴 ∪ 𝐴 = 𝑋
Law of contradiction
𝐴 ∩ 𝐴 = 𝜙
DeMorgan’s Law
|𝐴 ∩ 𝐵| = 𝐴 ∪ 𝐵; 𝐴 ∪ 𝐵 = 𝐴 ∩ 𝐵

Fuzzy Set Operations: Union
The union of fuzzy sets Type equation here.

References
Rajasekaran, S., & Pai, G. V. (2017). Neural Networks, Fuzzy Systems and Evolutionary Algorithms: Synthesis and
Applications. PHI Learning Pvt. Ltd..
Haykin, S. (2010). Neural Networks and Learning Machines, 3/E. Pearson Education India.
Sivanandam, S. N., & Deepa, S. N. (2007). Principles of soft computing. John Wiley & Sons.

Introduction to Fuzzy logic

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