Algebras of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 39

μ

Example 2.1.2 With X and in the last example, it is

, μ ( 0 . 6 ) ( , μ ( 0 . 7 ) ( μ( x ) = max ( min ( 0 . 6 x )), min ( 0 . 7 x )),

, μ ( 0 . 8 ) ( , μ ( 1 ) ( min ( 0 . 8 x )), min ( 1 x ))),

and, for instance,
μ( 1 ) = max ( min ( 0 . 6 , 1 ), min ( 0 . 7 , 1 ), min ( 0 . 8 , 1 ), min ( 1 , 0 )) = 0 . 8

μ( 4 ) = max ( min ( 0 . 6 , 1 ), min ( 0 . 7 , 1 ), min ( 0 . 8 , 1 ), min ( 1 , 1 )) = 1

etc.

Example 2.1.3 With X ={ 1 , 2 , 3 } ,take μ : X × X ₒ[ 0 , 1 ] , given by μ( i , j ) =

min ( i , j )

. This fuzzy set in X × X can be represented either by the matrix

⊛

3

⊞

1
3

1
3

1
3

⊝ ⊠ ,

1
3

2
3

2
3

1
3

2
3

1

or by the graph

2/3

1/3

1/3

2

1

1/3

2/3

1/3

2/3

1/3

3

1

2
3 ) , and

1
3 ) are respectively

μ ( 1 ) , μ ( μ (

Since the matrices of

⊛ ⊞ ⊛ ⊞ ⊛ ⊞

000
000
001

000
011
011

111
111
111

⊝ ⊠ , ⊝ ⊠ , ⊝ ⊠ ,

it results

⊛ ⊛ ⊛ ⊞ ⊞ ⊛ ⊛ ⊞ ⊞ ⊛ ⊛ ⊞ ⊞ ⊞

000
000
001

000
011
011

111
111
111

⊝ ⊝ ⊝ ⊠ ⊠ , ⊝ ⊝ ⊠ ⊠ , ⊝ ⊝ ⊠ ⊠ ⊠ 2 1

max min 1 , min

3 ,

min

3 ,

⊛
⊝

⊛
⊝

⊞
⊠ ,

⊛
⊝

⊞
⊠ ,

⊛
⊝

⊞
⊠

⊞
⊠ =

⊛
⊝

⊞
⊠ ,

1
3

1
3

1
3

1
3

1
3

1
3

000
0

000
000
001

2
3

2
3

1
3

1
3

1
3

1
3

2
3

2
3

= max

2
3

2
3

1
3

1
3

1
3

1
3

2
3

0

1

accordingly with the theorem of resolution.