l2 binomial operations
TRANSCRIPT
EE-646
Lecture-2
Binomial Operations on
Fuzzy Sets
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Norm Operators
• T-Norm Operator (T stands for triangular)
• is taken as min. (generally), i.e.
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( ) ( ) ( ),A B A BA B x x x x X
( ) ( ) min ( ), ( )A B A Bx x x x
T- Norm Operator
• However, can also be defined as:
i. Algebraic Product, •
ii. Bounded Product,
iii. Drastic Product,
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or
i. Algebraic Product
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,A B A BA B x x x x X
ii. Bounded Product
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max 0, ( ) ( ) 1 ,
A B
A B
A B x
x x x X
iii. Drastic Product
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( ), if ( ) 0
( ) ( ), if ( ) 0
0, otherwise
B A
A B A B
A B
x x
x x x
Note That
A B A B A B A B
Illustration on Board
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S-Norm Operator
• S-Norm Operator
• is taken as max. (generally), i.e.
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( ) ( ) ( ),A B A BA B x x x x X
( ) ( ) max ( ), ( )A B A Bx x x x
S- Norm Operator
• However, can also be defined as:
i. Algebraic Sum, +
ii. Bounded Sum,
iii. Drastic Sum,
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or
i. Algebraic Sum
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( ) ( ) ( ) ( ) ( ),A B A B A B
A B
x x x x x x X
ii. Bounded Sum
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( ) min ( ) ( ) , 1 ,A B A B
A B
x x x x X
iii. Drastic Sum
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( ), if ( ) 0
( ) ( ), if ( ) 0
1, otherwise
B A
A B A B
A B
x x
x x x
Note That
A B A B A B A B X
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Some More Operations
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1. Difference
or |
( ) max ( ) ( ) , 0
A B A B
A B A B
x x x
x X
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2. Absolute Difference
( ) ( ) ( ) ,A B A B
A B
x x x x X
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3. λ - Sum
( ) λ ( ) 1 λ ( ) ,
and λ [0, 1]
A B A B
A B
x x x
x X
λ
λ
+
+
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Order of Magnitude
A B A B A B
A B A B A B X
λ+
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Problem Task
Consider two fuzzy sets A and B as below:
Find
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0 0.5 0.3 0.7 0.9 0.2 0.4 0.6 0.9 0.4+ + + + & + + + +
1 2 3 4 5 1 2 3 4 5A B
(a) , (b) , (c) &
(d) | , (e)Verify De'Morgans Laws
(f) Algebraic Sum, and (g) Algebraic Product
C CA B A B A B
A B
Announcement
• From tomorrow onwards Thursday classes will be held in the first period
• Kindly make arrangements for the projector in advance
• Thank You!
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