Randolph diagram

A Randolph diagram (R-diagram) is a simple way to visualize logical expressions and combinations of sets. Randolph diagrams were created by mathematician John F. Randolph in 1965, during his tenure at the University of Arkansas.

Overview

Randolph diagrams can be interpreted most easily by defining each line as belonging to or relating to one logical statement or set. Any dot above the line indicates truth or inclusion and below the line indicates falsity or exclusion. Using this system, one can represent any combination of sets or logical statements using intersecting lines.

Though Venn diagrams are more commonly used to represent combinations of sets, Randolph diagrams have the advantage of being able to cleanly represent combinations of more than 3 sets. Venn diagrams require either extension into higher spatial dimensions or the use of more complicated shapes while Randolph diagrams evenly subdivide for every additional set. Here is a comparison between a Venn diagram and R-diagram for 5 sets of logical statements:

History

In his introductory paper on the subject, Cross-Examining Propositional Calculus and Set Operations, Randolph mentions that the first use of crosses and dots to represent logical relationships was introduced by W. S. McCulloch, a neurophysiologist and Randolph's contemporary. Randolph modified McCulloch's system with a new way of representing combinations and relationships of more than two logical statements or sets, namely subdividing each section of the R-diagram with a new diagonal line for each new element introduced. Randolph's paper suggests that his original notion was to use R-diagrams to represent logical relationships, and then expanded the idea to be applied to set theory as well.