Journal P.Jayasri, IJSRM IJSRM Academic Publisher 10.18535/ijsrm/v6i2.m02 Fuzzy Set Theory and Arithmetic Operations On Fuzzy N Umbers Definition: Fuzzy set Arithmetic Operations On Fuzzy Numbers 2018 06 02 4 7

The development of fuzzy set theory٫ since its introduction in 1965 has been dramatic. Thefuzzy set theory has pervaded almost all fields of study and its applications have percolated down to consumer goods level! Apart from this٫ it is being applied on a major scale in industries through intelligent robots for machine building ( cars٫ engines٫ turbines٫ ship٫ etc.) and controls and of course for military purposes. If X is a collection of objects denoted generally by X, then a fuzzy set A in X is a set of order pairs. ~ A={ X , µ~A( x)/ x ϵX } Where µ~A( x ) is called membership function or grade of membership.

Fuzzy set Fuzzy numbers Arithmetic operation Arithmetic intervals
Fuzzy Set Theory and Arithmetic Operations On Fuzzy N Umbers Introduction

The development of fuzzy set theory٫ since its introduction in 1965 has been dramatic. Thefuzzy set theory has pervaded almost all fields of study and its applications have percolated down to consumer goods level! Apart from this٫ it is being applied on a major scale in industries through intelligent robots for machine – building ( cars٫ engines٫ turbines٫ ship٫ etc.) and controls and of course for military purposes.

Keywords: Fuzzy set, Fuzzy numbers, Arithmetic operation, Arithmetic intervals.

Preliminaries Definition: Fuzzy set

If X is a collection of objects denoted generally by X, then a fuzzy set A in X is a set of order pairs. A~={X,µA~x/xϵX}

Where µA~x is called membership function or grade of membership.

Example:

Let X is a ten natural numbers

A ~

Arithmetic Operations On Fuzzy Numbers Definition:

Let A and B denote fuzzy numbers and let denote any of the four arithmetic operations. Then we define a fuzzy set on R, AB. by defining its cut , αAB as

αAB=αAαBαϵ

A*B = αϵ0,1αAB

Since, αAB is closed interval for each αϵ0,1 and A,B are fuzzy numbers are also a fuzzy numbers.

Definition:

Let * denote any of the four basic arithmetic operators and let A,B denote fuzzy numbers. Then we define a fuzzy set on R, A*B by the equation,

(A*B) (z) = minAx,ByzϵRz=xy

NOTE:

(A+B) (z) = minAx,ByZ=x+y

(AB) (z) = minAx,ByZ=xy

(A.B) (z) = minAx,ByZ=x.y

(A/B) (z) = minAx,ByZ=xy

Arithmetic Operation On Intervals

Let * denote any of the four arithmetic operations on closed intervals,

Subtraction

×Multiplication

Division

Then [ a , b ] * [ d , e ] = { f * g / afb ,dge} is a general property of all arithmetic operations on closed intervals accept that a,b/d,e is not defined when 0ϵd,e.

The result of an arithmetic operation on closed intervals is again a closed intervals.

l x α = A x n α

l x α α

Similarly, we can prove that,

y α ϵ α A

αA is closed interval

x α , y α ϵ α A

[ xα,yα αA

A is a fuzzy number.

Definition:

The four arithmetic operation on closed intervals are defined as followed.

[ a,b ] + [ d,e ] = [ a+d , b+c ]

[ a,b ] - [ d,e ] = [ a-d , b-c ]

[ a,b ] * [d,e ] = [ min ( ad ,ae ,bd ,be ) max ( ad, ae ,bd ,be ) ]

a,b/d,e= [a,b] * [1d,1e

m i n a d , a e , b d , b e m a x a d , a e , b d , b e

Example:
[ a,b] + [ d,e ] = [ a+d , b+c ]

[ 1,2 ] + [ 3,4 ] = [ 1+3, 2+4 ]

= [ 4 , 6 ]

[ a,b ] – [ d,e ] = [ a-e ,b-d ]

[ 1,2 ] – [ 3,4 ] = [1,2,3,4,2,3]

= [ -3 ,-1 ]

[ a,b ] * [ d,e ] = [ min ( ad,ae,bd,be ) , max ( ad,ae,bd,be ) ]

[ 1,2 ] * [ 3,4 ] = [ min ( 3,4,6,8) , max ( 3,4,6,8) ]

= [ 3,8 ]

[ a,b ] / [ d,e ] = [ min (<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"><mml:mfrac><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mi>m</mml:mi><mml:mi>a</mml:mi><mml:mi>x</mml:mi><mml:mfenced separators="|"><mml:mrow><mml:mfrac><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>d</mml:mi></mml:mrow></mml:mfrac><mml:mo>,</mml:mo><mml:mfrac><mml:mrow><mml:mi>b</mml:mi></mml:mrow><mml:mrow><mml:mi>e</mml:mi></mml:mrow></mml:mfrac></mml:mrow></mml:mfenced></mml:math>

[ 1,2 ] / [ 3,4 ] = [ min (13,14,23,24,max13,14,23,24

= [13,24

[ 1,2 ] / [ 3,4 ] = [ 13,12

Conclusion

In the paper discussed some result in fuzzy set theory and fuzzy arithmetic number. Here provide the fuzzy numbers as well as explain and example. This results obtained by using fuzzy arithmetic are applicable for the control system. In applied of fuzzy set theory the field of engineering has undoubtedly been leader. Fuzzy set theory is also becoming important in computer engineering.

Reference KlirGeorgeYuanBoBasic concepts and history of fuzzy set theory and fuzzy logicHandbook of Fuzzy ComputationIOP Publishing LtdBohlenderGerdKaufmannArnoldGuptaMadanMIntroduction to Fuzzy Arithmetic, Theory and Applications.Mathematics of Computation1986-octGrahamIanFuzzy set theory and its applications (2nd Edition)Fuzzy Sets and Systems1991-aug401402