T-norms and S-norms
Important set connectivesThis 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.
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
They have the following relationship:
Operators that are more general
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
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.