How did John Venn’s diagrams represent class relationships?

John Venn introduced his diagrams as a graphical method to represent relations of inclusion and exclusion between classes, or sets. These diagrams consist of two or three intersecting circles, with each circle representing a specific class labeled by an uppercase letter.12

To represent logical relations, Venn utilized two primary visual indicators:
* Shading: Used to indicate the nonexistence of members in a given class (an empty set).12
* Lowercase x: Used to indicate the existence of at least one member in a given class.12

Two-circle diagrams are used to represent categorical propositions, such as "All A are B" or "Some A are B," by shading or placing an x in specific intersections or regions of the circles. Three-circle diagrams are used to test the validity of categorical syllogisms. By diagramming the premises, one can determine if the conclusion is necessarily represented; if it is, the argument is considered valid. Venn developed this method in his work, Symbolic Logic, to support the algebraic logic of George Boole.12