T-Norm

A T- norm, often small t-norm also, is a mathematical function that has gained importance in the field of multivalent logics, especially in fuzzy logic. The term derives triangular norm from English to German standard triangle from, and due to the fact that a T - norm describes a triangle-like area in.

Properties

A T standard is defined on the unit interval [0,1]

And must have the following properties ( for the exact definition of these properties see table in T- norm and T- conorm at the end of this article):

  • Associativity: T (A, T ( B, C) ) = T (T (a, b ), c)
  • Commutativity T ( a, b) = T ( B, A)
  • Monotonicity: T (a, b ) ≤ T ( c, d ) if a ≤ b ≤ c, and d
  • 1 is a neutral element: T (a, 1 ) = a

The T- norm is used to provide a generalized conjunctive operator -valued logics. The above characteristics are equally common characteristics of such an operator: associativity and commutativity are self-evident. The monotony guarantees a certain regularity in the structure of definition and target quantity. The "1" as a neutral element allows for conjunctions, the result depends only on one operand.

These properties are used in conjunction with fuzzy sets to reproduce the intersection operation.

T- conorms

Complementary to T- norms are t- conorms ( S- norms also called ) are used. With the help of De Morgan 's laws can be namely on the basis of a t- norm, which delivers conjunction or intersection, and negation to derive the disjunction or the union operation.

Common T- norms and T- conorms

The indicated T are conorms = 1-x for the corresponding T standard dual, that is about the De Morgan 's associated respectively with respect to the standard negation N (x) laws. In other involutive negations arise in general, other T- conorms.

The former is used because of their simplicity and their attributes below the most common. The 3 T standard, as well as their T- conorm come from the theory of probability. Furthermore, the following relationships apply:

That that the drastic T standard (T-1 ) is the smallest, and the minimum -T standard is the largest. The opposite is true for the T- conorm. T ( A, B ) or ⊥ (a, b ) stands for any T- T standard or any conorm.

Correlations between T- norm and T- conorm

Due to the already mentioned De Morgan 's laws, the following relationships result:

The following conditions are required in order to function as a T- norm and T - conorm applies:

Many-valued logic Monotonicity of entailment Logical conjunction Involution (mathematics)
699145
de