The discussion of set theory would not be complete without the inclusion of a useful tool known as Venn diagram. Venn diagram is useful in visualizing set operations, in which logical sets are presented pictorially. It was introduced by John Venn in 1880 in a paper entitled,
The circle inside a Venn diagram, at the time of Venn, was referred to as Euler's circle was later replaced with Jordan curve, a more apt terminology for the circles inside the universe of discourse.
Disclaimer: this is a summary of section 4.4 from the book A Gentle Introduction to the Art of Mathematics: by Joe Fields, the content apart from rephrasing is identical, most of the equations are screenshots of the book and the same examples are treated.
Venn Diagram
Venn diagrams take advantage of an obvious but important property of closed curves drawn on the plane. They divide the points in the plane into two sets, those that are inside the curve and those that are outside!
This seemingly obvious statement is known as the Jordan curve theorem and actually requires some details. Jordan curve theorem is one of those statements that hardly seems like it needs a proof, but nevertheless, the proof of this statement is probably the best-remembered work of the famous French mathematician Camille Jordan.
an example of a Venn diagram
In a Venn diagram, the universe of discourse is normally drawn as a rectangular region inside of which all the action occurs. Each set in a Venn diagram is depicted by drawing a simple closed curve, typically a circle, but not necessarily!
Here is a small example of a finite universe
our universe of discourse
We can create sets using a Venn diagram.
the set of cartoon characters C and the set of horses H are disjoint with the Jordan curve
Another element was added to our universe in order to dispel the notion that the sets are that of cartoon characters and horses.
Disjoint and Containment
One can use Venn diagram to represent disjoint sets and, if one set in known to be contained in the other, to represent containment.
Set Operations
There is a connection between the regions in a Venn diagram to the set operations we’ve discussed on set operations.
depiction of set operations inside the Jordan curve