## 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 { *f _{i}*(X

_{i}) } , where Y is the outcome, X

_{i}is the i-th condition, and

*f*(X

_{i}_{i}) is the i-th ceiling line. For example when two conditions have ceiling lines

*f*(X

_{1}_{1}) = X

_{1}, and

*f*(X

_{2}_{2}) =X

_{2}then the ceiling surface is defined by Y

_{c}= min {X

_{1}, X

_{2}}, see figure, and the necessary AND configuration by Y ≤ min {X

_{1}, X

_{2}}.

**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).