# Optimal Classification/Rypka Method/Equations/Separatory/Elements

## Element Related EquationsEdit

### Order of elementsEdit

The elements are arranged in descending order according to their truth table value, i.e., the value calculated as the sum of each characteristic's logic state value times the highest value of logic raised to the power of the order of the characteristic.[1] The element truth table value allows elements to be sorted and identified as unique or equivalent and the bounded class identified as a set or multiset.

${\displaystyle e_{i}=\sum _{j=0}^{C}\left[v_{i,j}V^{(C-j)}\right]}$ , where:
• ei is the element truth table value in the group,
• V is the highest value of logic in the group,
• v is the value of logic of each characteristic in the group,
• j is the jth characteristic index, where:
j = 0..C and where:
• C is the number of characteristics in the group,
• i is the ith element index, where:
i = 0..G and where:
• G is the number of elements in the bounded class.

### Maximum number of pairs of elements to separateEdit

Maximum number of pairs of elements to separate refers to triangularization of the matrix to permit comparison of each element with every other element to determine the number of pairs that are separable or disjoint. Pairs are separable or disjoint whenever the logic values of the elements that make up a pair are different. In theory, therefore the maximum possible number of pairs that can be separated is determined by the following equation:[2]

${\displaystyle p_{max}={\frac {\left[{G(G-1)}\right]}{2}}}$ , where:[3]
• pmax is the maximum number of pairs to separate, and
• G is the number of elements in the bounded class.

## NotesEdit

1. See Number Systems - essentially using the characteristic values to compute a network or memory address, followed by sorting.
2. See page 176 Table XI of the primary reference.
3. Triangular Number