# Intersection-union systems

## Abstract

In two recent papers the authors introduced some new approaches to the problem of modeling the phenomena underlying immunological reaction tests. These approaches allow one to easily construct the best possible models for a given amount of immunological test data. In the first paper the authors used a purely Boolean approach, i.e., it was assumed that the experimenter signified whether or not a reaction occurred in a given test. In the second paper the authors assumed that the relative strengths of the reactions were available as data for the modeling process. They showed that in this case strictly better models could be constructed. This paper generalizes the approaches taken in the first two papers and provides a unified approach to this whole subject. Many of the results, e.g., the ability to construct the best model, of the first two papers hold in this more general setting. Moreover, this generalization allows one to assess the tradeoffs involved in using data on the relative strengths of reactions. In particular, we see that using relative strengths is equivalent to using an additional intersection factor in a strictly Boolean approach. This intersection factor it turns out, can be obtained experimentally by using elution in addition to the absorption involved in the first two papers. Finally, the duality between fragments and cofragments becomes apparent using this approach. © 1983.