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Fuzzy Logic
B. Logic Operations
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London/9.0YorkNew/8.0Chicago/5.0LA/0.0
LA
Conventional or crisp sets are binary. An element
either belongs to the set or doesn't.
Fuzzy sets, on the other hand, have grades ofmemberships. The
set of cities `far' from Los Angeles
is an example.
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The set,B, of numbersnear to two is
or...
2)2(
1
)(
xxB
|2|
)(
x
B ex0 1 2 3 x
B(x)
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A fuzzy set,A, is said to be a subset ofB if
e.g.B = far andA=very far.
For example...
)()( xx BA
)()( 2 xxBA
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Crisp membership functions are either one orzero.
e.g. Numbers greater than 10.
A ={x |x>10}
1
x
10
A(x)
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Criteria for fuzzy and, or, and complement
Must meet crisp boundary conditions
Commutative
Associative
Idempotent
Monotonic
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Example Fuzzy Sets to Aggregate...
A = {x |x is nearan integer}
B = {x |x is close to 2}
0 1 2 3 x
A(x)
0 1 2 3 x
B(x)
1
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)](),([max)( A xxx BBA
Fuzzy Union (logic or)
Meets crisp boundary conditions
Commutative
Associative
Idempotent
Monotonic
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0 1 2 3 x
A+B (x)
A ORB =A+B ={x | (x is nearan integer) OR (x is close to 2)}
= MAX [A(x), B(x)]
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)](),([min)( A xxx BBA
Fuzzy Intersection (logic and)
Meets crisp boundary conditions
Commutative
Associative
Idempotent
Monotonic
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A ANDB =AB ={x | (x is nearan integer) AND (x is close to
2)}
= MIN [A(x), B(x)]
0 1 2 3 x
AB(x)
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The complement of a fuzzy set has a membership function...
)(1)( xxAA
0 1 2 3 x
1
)(xA
complement of A ={x |x is not nearan integer}
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)(1)( xxAA
Meets crisp boundary conditions
Two Negatives Should Make a Positive
Monotonic
Law of Excluded Middle: NO!
Law of Contradiction: NO!
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Min-Max fuzzy logic has intersection distributive over
union...
)()( )()()( xx CABACBA
since
min[A,max(B,C) ]=min[ max(A,B), max(A,C) ]
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Min-Max fuzzy logic has union distributive over
intersection...
)()( )()()( xx CABACBA
since
max[A,min(B,C) ]= max[ min(A,B), min(A,C) ]
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Min-Max fuzzy logic obeys DeMorgans Law #1...
since
1 - min(B,C)= max[ (1-A), (1-B)]
)()( xx CBCB
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Min-Max fuzzy logic obeys DeMorgans Law #2...
since
1 - max(B,C)= min[(1-A), (1-B)]
)()( xx CBCB
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Min-Max fuzzy logic fails The Law of Excluded Middle.
since
min(A
,1-A
) 0
Thus, (the set of numbers close to 2) AND (the set of
numbers not close to 2) null set
AA
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U AA
Min-Max fuzzy logic fails theThe Law of Contradiction.
since
max(A
,1-A
) 1
Thus, (the set of numbers close to 2) OR (the set of
numbers not close to 2) universal set
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The intersection and union operations can also be used to
assign
memberships on the Cartesian product of two sets.
Consider, as an example, the fuzzy membership of a set, G,
of
liquids that taste goodand the set, LA, of cities close to
LosAngeles
JuiceGrape/9.0
JuiceRadish/5.0
WaterSwamp/0.0
G
London/9.0YorkNew/8.0
Chicago/5.0LA/0.0
LA
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We form the set...
E = G LA= liquids that taste goodAND cities that
are close to LA
The following table results...
LosAngeles(0.0) Chicago (0.5) New York (0.8) London(0.9)
Swamp Water (0.0) 0.00 0.00 0.00 0.00
Radish Juice (0.5) 0.00 0.50 0.50 0.50
Grape Juice (0.9) 0.00 0.50 0.80 0.90
