## T-norms and S-norms## Important set connectivesThis page deals with set connectives.For better reading I write for μ_{A}(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
Look here for the diagrams of the T- 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. |

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