T-norms and S-norms

Important set connectives

This page deals with set connectives.
For better reading I write for μA(x) (= the grade of membership of x in A): A.
So A is the fuzzy set A.
Now let's take two fuzzy sets A and B.

The minimum operator is the well known operator to model intersection. It is the red line in the picture below.

This minimum operator is one of the "triangular norms".
With the T-norms there are the T-conorms, also called the S-norms.
They model union. The maximum operator is an S-norm.
It is the blue line in the picture above.
Apart from that I mention the algebraic sum of A and B: A + B

T-norms and S-norms are logical duals and can be computed from each other by:

t(A,B) = 1 - s(1-A, 1-B)

A few well known couples of T-norms and S-norms

Drastic product
Drastic sum
tw(A,B) = min (A,B) if max (A,B) = 1, else = 0
sw(A,B) = max (A,B) if min (A,B) = 0, else = 1
Bounded difference
Bounded sum
t1(A,B) = max (0, A + B - 1)
s1(A,B) = min (1, A + B)
Einstein product
Einstein sum
t1.5(A,B) = (AB/(2 - [ A + B - AB])
s1.5(A,B) = (A + B)/ (1 + AB)
Algebraic product
Probabilistic sum
t2(A,B) = AB
s2(A,B) = A + B - AB
Hamacher product
Hamacher sum
t2.5(A,B) = (AB)/(A + B - AB)
s2.5(A,B) = (A + B - 2AB)/(1 - AB)
t3(A,B) = min (A,B)
s3(A,B) = max (A,B)

Look here for the diagrams of the T- and S-norms.

They have the following relationship:

tw <= t1 <= t1.5 <= t2 <= t2.5 <= t3
s3 <= s2.5 <= s2 <= s1.5 <= s1 <= sw

Operators that are more general

FANDFuzzy ANDFAND(A,B) = γ*min(A,B) + 0.5(1-γ)(A + B)    γ Î [0,1]
FORFuzzy ORFOR(A,B) = γ*max(A,B) + 0.5(1-γ)(A + B)      γ Î [0,1]

Even more general are convex combinations of T-norms with their S-norms. For example the convex combination of the minimum and the maximum operator. Or of the algebraic product and the probabilistic sum.

Some convex combinations of T-norms with their S-norms

Convex combination
of min and max
Comb(A,B)= γ*min(A,B)+(1-γ)*max(A,B)    γ Î [0,1]
Convex combination
of algebraic product and
probabilistic sum
Comb(A,B)= γ*(AB)+(1- γ)*(A+B-AB)       γ Î [0,1]

So it's obvious that there are a lot of operators to model union and intersection.
This is one of the features of fuzzy logic, that makes it so well suited to handel the vagueness of the real world.

Source: T.Tilly, "FUZZY LOGIC, theorie, praktijk, hard-en software", Kluwer Techniek, ELEKTRO/ELEKTRONICA.

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