Mathematical Description

What is the general mathematical description of a necessary condition?

What is the mathematical description of a necessary AND configuration?

What is the mathematical description of a necessary OR configuration?

 

What is the general mathematical description of a necessary condition?

The general mathematical description of a necessary condition is Y ≤ f(X), where Y is the outcome, X is the condition, and f(X) is the ceiling. With one condition f(X) is a ceiling line (line on the data), with two conditions f(X) is a ceiling surface (blanket on the data), and with more than two conditions f(X) is a multidimensional surface.

What is the mathematical description of a necessary AND configuration?

The mathematical description of a necessary AND configuration with several conditions is: Y ≤  min { fi(Xi) } , where Y is the outcome, Xi  is the i-th condition, and fi(Xi)  is the i-th ceiling line. For example when two conditions have ceiling lines f1(X1) = X1, and f2(X2) =X2 then the ceiling surface is defined by Yc = min {X1, X2}, see figure, and the necessary AND configuration by Y ≤ min {X1, X2}.

Mathematical description of necessary AND configuration

What is the mathematical description of a necessary OR configuration?

The mathematical description of a necessary OR configuration with several conditions is Y ≤ f(X), where Y is the outcome, X is a vector with the conditions, and f(X) is a joint multidimensional ceiling. For example when two conditions have a joint ceiling f(X1,X2) = (X1 + X2) then the ceiling surface is defined by Y = X1 + X2, see figure, and the necessary OR configuration by Y ≤ (X1 + X2).

Mathematical description of a necessary OR configuration