Introduction To Fuzzy Set

Introduction: -The word "fuzzy" means "vagueness". Fuzziness occurs when the boundary of a piece of information is  not  clear-cut.  Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the  classical  notion of  set.   Classical  set theory allows  the  membership  of  the  elements  in  the se in  binary  terms,  a  bivalent  condition -  an element  either  belongs  or does  not  belong  to  the  set.

Fuzzy  set theory  permits  the gradual  assessment  of  the  membership of elements in a set, described with  the aid of a membership function valued  in  the  real  unit  interval [0, 1].

Example:  

Words  like  young,  tall,  good,  or  high  are  fuzzy. 

−  There  is  no  single  quantitative value  which defines  the term young. 

−  For  some people,  age 25 is young, and  for others, age 35  is  young. 

−  The  concept  young  has  no  clean  boundary. 

−  Age 1  is  definitely  young  and  age 100  is  definitely  not  young.

−  Age 35  has some possibility of being young and usually depends  on  the  context  in  which  it  is  being  considered.

In  real  world,  there  exists  much  fuzzy  knowledge;   Knowledge that is  vague, imprecise, uncertain, ambiguous, inexact, or probabilistic in nature.  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. our systems should also be able to cope with unreliable and incomplete information and give expert opinions. Fuzzy sets   have  been able  provide  solutions  to  many  real  world  problems. Fuzzy  Set  theory  is  an  extension  of  classical  set  theory  where  elements have  degrees  of  membership.

Fuzzy Set Theory

Fuzzy  set  theory  is  an  extension  of  classical  set  theory  where elements  have  varying  degrees  of  membership.  A  logic  based on the two truth values,  True and  False, is sometimes inadequate when describing human reasoning.  Fuzzy  logic  uses the whole interval between 0 (false) and 1 (true) to describe human reasoning.  

−  A  Fuzzy Set  is any  set  that  allows  its  members  to  have  different degree  of  membership,  called  membership function,  in the interval [0 , 1].

-  The  degree of membership  or  truth is not same as probability;  fuzzy truth is not likelihood of some event or condition.  Fuzzy truth represents membership in vaguely defined sets.

−  Fuzzy  logic  is derived from fuzzy set theory dealing  with  reasoning that is approximate rather than precisely deduced from classical predicate logic.

−  Fuzzy logic is capable of handling inherently imprecise concepts.

−  Fuzzy logic allows in linguistic form the set membership values to imprecise concepts like "slightly", "quite" and "very".

−  Fuzzy set theory defines Fuzzy Operators on Fuzzy Sets.