Fuzzy Arithmetic - Fuzzy Logic: An Introductory Course for Engineering Students - page 147

(μ ˃) ʱ = μ ʱ ˃ ʱ ,

with the particular supposition that when ∗=: , it should be required that 0 ∈ μ ʱ for

all ʱ ∈ ( 0 , 1 ] . Notice that it also follows

μ ˃ = ʱ ∧ (μ ∗ ˃) ʱ = ʱ ∧[ μ ʱ ˃ ʱ ] .

ʱ ∈[ 0 , 1 ] ʱ ∈[ 0 , 1 ]

It is for this equality that the before hand computations were made. For example,
with

⊧
⊨

⊧
⊨

x + 1 x − 1

, if t ∈ ( − 1 , 1 ) , if t ∈ ( 1 , 3 ]

2 2

3 − x − x

2

5

(μ 1 )( x ) = , if t ∈ ( 1 , 3 ) ,(μ 3 )( x ) = , if t ∈ ( 3 , 5 )

⊩

2

⊩

0 , otherwise 0 , otherwise

the corresponding ʱ -cuts are

(μ 1 ) ʱ =[ 2 ʱ − 1 , 3 − 2 ʱ ] ,(μ 3 ) ʱ =[ 2 ʱ + 1 , 5 − 2 ʱ ] ,

with which

[ μ 1 ↕ μ 3 ] ʱ =[ 4 ʱ, 8 − 4 ʱ ] ,for ʱ ∈ ( 0 , 1 ] .

Hence,

⊧
⊨

x

4 ,

if t ∈ ( 0 , 4 ]

8 − x

(μ 1 ↕ μ 3 )( t ) = , if t ∈ ( 4 , 8 ]= μ 4 ( t )

⊩

4

0 , otherwise

6.3 A Note on the Lattice of Fuzzy Numbers

As it is well known, ( R , min , max ) is a distributive lattice that come from the totally

ordered set ( R , ≤ ). The order ≤ is definable from the lattice operations min , max by

a ≤ b ⃔ a = min ( a , b ) ⃔ b = max ( a , b ).

In addition, with a a , b )) , and max = 1 − a ,itis min ( a , b ) = ( max ( ( a , b ) =

a , b )) . Let's extend these operations to the set R ∗ of all fuzzy numbers, by ( min (