Prerequisite : Fuzzy Logic | Introduction
In this post, we will discuss classical sets and fuzzy sets, their properties and operations that can be applied on them.
Set: A set is defined as a collection of objects, which share certain characteristics.
- Classical set is a collection of distinct objects. For example, a set of students passing grades.
- Each individual entity in a set is called a member or an element of the set.
- The classical set is defined in such a way that the universe of discourse is spitted into two groups members and non-members. Hence, In case classical sets, no partial membership exists.
- Let A is a given set. The membership function can be use to define a set A is given by:
- Operations on classical sets: For two sets A and B and Universe X:
This operation is also called logical OR.
This operation is also called logical AND.
- Fuzzy set is a set having degrees of membership between 1 and 0. Fuzzy sets are represented with tilde character(~). For example, Number of cars following traffic signals at a particular time out of all cars present will have membership value between [0,1].
- Partial membership exists when member of one fuzzy set can also be a part of other fuzzy sets in the same universe.
- The degree of membership or truth is not same as probability, fuzzy truth represents membership in vaguely defined sets.
- A fuzzy set A~ in the universe of discourse, U, can be defined as a set of ordered pairs and it is given by
- When the universe of discourse, U, is discrete and finite, fuzzy set A~ is given by
- Fuzzy sets also satisfy every property of classical sets.
- Common Operations on fuzzy sets: Given two Fuzzy sets A~ and B~
where “n” is a finite value.
- Union : Fuzzy set C~ is union of Fuzzy sets A~ and B~ :
- Intersection: Fuzzy set D~ is intersection of Fuzzy sets A~ and B~ :
- Complement: Fuzzy set E~ is complement of Fuzzy set A~ :
- Algebraic sum:
- Algebraic product:
- Bounded sum:
- Bounded difference:
Attention reader! Don’t stop learning now. Get hold of all the important CS Theory concepts for SDE interviews with the CS Theory Course at a student-friendly price and become industry ready.
- Fuzzy Logic | Introduction
- Classical Computing vs Quantum Computing
- Common Operations on Fuzzy Set with Example and Code
- Comparison Between Mamdani and Sugeno Fuzzy Inference System
- ML | Fuzzy Clustering
- Perceptron Algorithm for Logic Gate with 3-bit Binary Input
- Frequent Item set in Data set (Association Rule Mining)
- Proto Van Emde Boas Tree | Set 3 | Insertion and isMember Query
- Van Emde Boas Tree | Set 1 | Basics and Construction
- Proto Van Emde Boas Tree | Set 6 | Query : Successor and Predecessor
- Introduction to Artificial Neutral Networks | Set 1
- Introduction to Artificial Neural Network | Set 2
- Lowest Common Ancestor in a Binary Tree | Set 3 (Using RMQ)
- Longest Palindromic Substring using Palindromic Tree | Set 3
- Introduction to ANN (Artificial Neural Networks) | Set 3 (Hybrid Systems)
- Introduction to ANN | Set 4 (Network Architectures)
- Mandelbrot Fractal Set visualization in Python
- How to use Vision API from Google Cloud | Set-2
- Data Mining | Set 2
- Proto Van Emde Boas Trees | Set 4 | Deletion
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.