Algebras of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 48

that corresponds to “solve” the equations with fuzzy sets,

μ · μ = μ 0 , μ + μ = μ 1 ,

that is, to find for which intersections , these equations do hold.

Of course, they do not hold in all cases, for example, with N

· and which unions +

= 1 − id,

• If ·= min, it is not always min (μ( x ), 1 − μ( x )) = 0 ,

• If += max, it is not always max (μ( x ), 1 − μ( x )) = 1 ,

• If ·= W ,( W ( a , b ) = max ( 0 , a + b − 1 )) it is W ( a , 1 − a ) = max ( 0 , a + 1 −

a − 1 )) = 0, and W (μ( x ), 1 − μ( x ))) = 0 for all x in X

W ∗ ,( W ∗ ( ,itis W ∗ (μ(

• If += a , b ) = min ( 1 , a + b )) x ), 1 − μ( x ))) = 1 for all x

in X .

That is, there are algebras of fuzzy sets where this forms of non-contradiction
or excluded-middle hold , and algebras where this principles do not jointly hold .
In the algebras with the triplet ( min , max , 1 − id ) do not hold both principles, in

the algebras with it holds the principle of non-contradiction but

not that of excluded-middle, in the algebras with

( W , max , 1 − id )

W ∗ ,

it holds the
excluded-middle but not the principle of non-contradiction, and in the algebras with
(

( min , 1 − id )

W ∗ , W , 1 − id ) both principles hold.

X

μ · μ

Remark 2.2.15 Let us show that with μ ∈{ 0 , 1 } it is always = μ 0 and

μ + μ = μ 1 .If

X , denote A

μ ∈{ , } ={ ∈ ; μ( ) = } μ = μ A

0 1 x X x 1 . Obviously,

μ = μ A c , hence
• (μ · μ ) = μ A · μ A c

and

= μ A ∩ A c = μ ∅ = μ 0

• (μ + μ ) = μ A + μ A c = μ A ∪ A c = μ X = μ 1 .

Remark 2.2.16 Results in Theorems 2.2.13 and 2.2.14 challenge the usual statement
that in fuzzy sets the basic principles of Non-contradiction and Excluded-middle fail.
A statement that could conduct to believe that fuzzy set algebras are not properly
grounded in a solid ground.
The fact is, notwithstanding, that these two principles were established before
the current ways of considering the problems of logic and, of course, before the
nomenclature of set theory. In set theory (or Boolean algebras), A A c ∩ = ∅ and

A c A c

c

A ∩ ↂ ( A ∩ ) are equivalent formulas since, as it was said, it is

B c B = ∅ ⃔ B ↂ ,

an equivalence only verified in the setting of ortholattices (of which Boolean algebras
are a particular case), but that does not hold on weaker algebraic structures like it
is, for example, the case of the above defined algebras of fuzzy sets. Let us call
'restricted' the principles stated by

μ · μ = μ 0 and μ + μ = μ 1 .