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CHAPTER 2

OPERATIONS ON FUZZY SETS

. Fuzzy Complement

. Fuzzy Union

. Fuzzy Intersection

. Combinations of Operations

. General Aggregation Operations . Fuzzy Complement Def: A complement of a fuzzy set A is specified by a function ]1,0[]1,0[: c , which assigns a value ( ))(xc A to each membership grade )(xA . Axiom: The function of a fuzzy complement must satisfy at least the following

two requirements:

c1.: 1)0( =c and 0)1( =c (boundary conditions). c2.: For all ]1,0[, ba , if ba < , then )()( bcac (c is monotonic

nonincreasing).

Remark: 1. All functions that satisfy axioms c1 and c2 form the most general

class of fuzzy complements. 2. Axioms c1 and c2 are called the axiomatic skeleton for fuzzy

complements. 3. Axiom c1 is the ordinary complement for crisp sets. Example: The threshold-type complement

>=

,for 0,for 1

)(tata

ac

where )1,0[ and ]1,0[ ta ; t is called the threshold of c. Axiom: In practical significance, it is desirable to consider various additional

requirements such as: c3. c is a continuous function. c4. c involutive, i.e. ( ) aacc =)( for all ]1,0[a .

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Example: An example that satisfies axioms c1 to c3

( )aac cos121)( +=

Examples of involutive fuzzy complement: 1. The sugeno class:

a

aac +=

11)( ,

where ),1( .

2. The yager class:

( ) www aac 11)( = , where w ( , )0 . Note: 1) For .1)( ,0 aac == 2) For .1)( ,1 aacw w ==

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. Fuzzy Union Def: The union of two fuzzy sets A and B is specified by a function of

the form: ]1,0[]1,0[]1,0[: . Formally, [ ])(),()( xxx BABA = . Axioms: The function of a fuzzy union must satisfy at least the following

axioms (the axiomatic skeleton for fuzzy set unions):

u1. 0)0,0( =u ; 1)1,1()0,1()1,0( === uuu (boundary conditions).

u2. ),(),( abubau = (commutative).

u3. If aa and bb , then ),(),( baubau (monotonic). u4. ( ) ( )),(,),,( cbuaucbauu = (associative).

Axiom: It is often desirable to restrict the class of fuzzy union by considering

various additional such as: u5. u is a continuous function. u6. aaau =),( , i.e. u is idempotent.

Example: The Yager class union satisfies u1 to u5. ( )[ ]wwww babau 1,1min),( += where ),0( w . Property: ,1=w ),1min(),(1 babau += 2=w , ( )222 ,1min),( babau += =w , ),max(),( babau =

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. Fuzzy Intersection Def: The intersection of two fuzzy sets A and B is specific by a

function: ]1,0[]1,0[]1,0[: i . Formally, [ ])(),()( xxx BABA = . Axioms: The function of fuzzy intersection must satisfy the following axioms

(the axiomatic skeleton for fuzzy set intersections):

i1. ;1)1,1( =i 0)0,0()0,1()1,0( === iii ; (boundary conditions). i2. ),(),( abibai = ; (commutative) i3. If aa and bb , then ),(),( baibai (monotonic). i4. ( ) ( )),(,),,( cbiaicbaii = (associative).

. The most important additional requirements:

i5. i is a continuous function. i6. aaai =),( , i.e. i is idempotent.

Example: The Yager class union satisfies i1 to i4: ( )[ ]wwww babai 1)1()1(,1min1),( +=

where ),0( w . Property: ,1=w )2,1min(1),(1 babai = 2=w , ( )222 )1()1(,1min1),( babai += =w , ),min(),( babai =

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. COMBINATION OF OPERATORS: Theorem: ),(),(),max( max baubauba , where

==

= otherwise. 1,0 when,0 when

),(max abba

bau

proof: 1) Using associativity (u4) ( ) ( )0),0,()0,0(, auuuau = By applying the boundary conditions (u1) ( )0),0,()0,( auuau = so aau =)0,( By monotonicity of u (u3) aaubau = )0,(),( By employing commutativity (u2) bbuabubau == )0,(),(),( Hence ),max(),( babau .

2) When 1. 0=b , then ),(),( max bauabau == 2. 0=a , then ),(),( max baubbau == 3. ]1,0(, ba , then ),(1),1()1,(),( max baubuaubau === From 1, 2, 3, ),(),(max baubau . Theorem: ),min(),(),(min babaibai where

==

= otherwise. 0,1 when,1 when

),(min abba

bai

proof: Similar to the previous theorem 1) Basing on the associativity (u4) ( ) ( )1),1,()1,1(, aiiiai = Using the boundary condition (u1) ( )1),1,()1,( aiiai = so aai =)1,( By monotonicity (u3) aaibai = )1,(),(

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With the commutativity (u2) bbiabibai == )1,(),(),( 2) Left as an exercise! Property:

1. The max operation is the lower bound of function u, i.e. the largest

union. 2. The min operation is the upper bound of function i, i.e. the weakest

intersection. 3. Of all possible pairs of fuzzy sets unions and intersections, the max and

min functions are closest to each other (an extreme pair), i.e. ),(),(),min(),max( baibaubababa =

Proposition: The max and min functions together with the aac =1)(

complement satisfy the DeMorgan's laws, i.e. )1,1min(1),max( baba = )1,1max(1),min( baba = Proof: See blackboard!

Proposition: The operations umax and umin is another extreme pair in the

sense that ),(),(),(),( minmax baibaubaibau The DeMorgan laws are also satisfied under the complement

operation aac =1)( . )1,1(1),( minmax baibau = )1,1(1),( maxmin baubai = Proof: An exercise! . The full scope of fuzzy aggregation operation: Dombi Dombi 0 0 Schweizer/Sklar Schweizer/Sklar - p p - Yager Yager 0 w w 0 Generalized means - mini i min max u maxu Intersection Average Union

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Theorem: ),max(),( babau = and ),min(),( babai = are the only continuous

and idempotent fuzzy set union and fuzzy set intersection. Proof: a) The fuzzy set union: By associativity ( ) ( )baauubauau ),,(),(, = The idempotency leads to(u6) ( ) ),(),(, baubauau = Similarly, ( ) ( ) ),(),(,),,( baubbuaubbauu == Hence ( ) ( )bbauubauau ),,(),(, = (*) When a = b, (*) is trival. W.l.o.g., Let a < b. 1) Assume =),( bau , where a and b .

Then (*) becomes ),(),( buau = (**) Since u is continuous (u5) and monotonic nondecreasing (u3) with

=),0(u and 1),1( =u

),(),(]1,0[, buauba

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. About the fuzzy set operations max and min The structure of { } ,,),(~ XP satisfies 1. Involution AA = 2. Commutativity ABBA = , ABBA = 3. Associativity )()( CBACBA = )()( CBACBA = 4. Distributivity 5. Idempotence 6. Absorption 7. Absorption by X and . 8. Identity 9. DeMorgan laws But violates a) Law of contradiction AA b) Law of excluded middle XAA Remark on violations: 1. Since a fuzzy set A has no sharp boundary, and neither has A , A and

A would overlap, so AA . However, the overlap is always limited, since

( )21)(),(min)( = xuxuxu AAAA

2. Also, AA does not cover X, but

( )21)(),(max)( = xuxuxu AAAA

. Alternative Operators on )(~ XP 1. Probabilistic operator Def: Intersection { }BAuXBA = , , )()()( xuxuxu BABA = , Xx Union { }BAuXBA +=+ , , )()()()()( xuxuxuxuxu BABABA +=+ , Xx Remark: 1) Probabilistic union and intersection can be deduced by the Dubois

and Prade class of fuzzy set operation, since as = 1,

a) abbaabbaba

baabba +=+=+

10

),1,1max()1,,min(

b) 1.

),,max(ba

baba =

2) { }+ ,,),(~ XP satisfies only Commutativity associativity identity DeMorgan's law =A , XXA =+ (absorption by and X)

0.5

1

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2. Bold operator Def: Intersection { }BAXBA = , ( )1)()(,0max)( += xxx BABA , Xx Union { }BAXBA = , ( ))()(,1min)( xxx BABA += , Xx Remark: 1) Bold union and intersection can be deduced by the Yager class

operator, since as w = 1, a). ( ) ),1min()(,1min 1 baba www +=+ b). ( )[ ]www ba 1)1()1(,1min1 +

( ))1,0max(

)1()1(,1min1+=

+=ba

ba

2) { } ,,),(~ XP satisfies only Commutativity associativity identity DeMorgan's law =A , XXA = (absorption by and X) XAA = , = AA (law of exclude middle)

. Comparison on { } ,,),(~ XP , { }+ ,,),(~ XP , { } ,,),(~ XP 1. In general,

BABABA

BABABA +

2. ( ) )()(1)()(,0max xxxx BABA +

( ) ( ))(),(max)(),(min xxxx BABA )()()()( xxxx BABA + ( ))()(,1min xx BA + . About the standard operators

1> Union { }BAXBA = , { })(),(max)( xxx BAXxBA = 2> Intersection { }BAXBA = , { })(),(min)( xxx BAXxBA = 3> Complement { }AXA ,= Xxxx AA = ),(1)(

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Example:

Ages Infant Adult Young Old5 0 0 1 010 0 0 1 0 20 0 .8 .8 .1 30 0 1 .5 .2 40 0 1 .2 .4 50 0 1 .1 .6 60 0 1 0 .8 70 0 1 0 1 80 0 1 0 1

801701608.506.

404.305.208.10151OldYoung++++

++++=

501.402.302.201.OldYoung +++= Note: When we take {0,1} instead of [0,1], then

1. { }{ })(),(max)(, xxxXBA BAXxBA == ={ }BxAxx or 2. { }{ })(),(min)(, xxxXBA BAXxBA == ={ }BxAxx and 3. { }AxXxxA = and

Comment: 1) From the note, they perform identically to the corresponding crisp

set operators when {0,1} is instead of [0,1]. 2) The max and min operations are additional significant in that they

constitute the only fuzzy union and intersection operators that are continuous and idempotent.

3) Both of these operations are very simple. 4) They are good generalizations of the classical crisp set operators. 5) A lot of applications are baseing on this pair of operations. ____________________________________________________________