By Bob Pitlak
The Venn Diagram is a graphical construction that
shows all the possible logical relationships between a
finite number of sets, or collections of similar things.
They are named after John Venn, a logician who
introduced them around 1880 and included them in
his book, Symbolic Logic, which he published
around 1881.
Venn Diagrams are useful in many areas
including set theory, logic, probability, statistics
and computer science.
Venn Diagrams are similar to the Euler Diagrams
introduced by Swiss mathematician Leonard Euler
in the 18th Century. However, true Venn Diagrams
contain “zones” that represent all possible logical
relationships, even if those zones contain a zero
population.
So, for example, a Venn Diagram of the set of “All
dogs” and the set of “All cats” would contain a region
to accommodate dogs that are also cats, even though
none exist. An Euler Diagram would eliminate such a
region.
We define our universe as “Animals”
With a sub-set of “small & Hairy”
A
B
We define a second sub-set as “web feet”
A
B
We continue to populate the sets…
A
B
Eventually, we may find something that belongs
to both sets.
A
B
We show this by overlapping the two sets, and
placing things that belong to both in the overlap
When all of a collection of things belongs to the same
sub-set, it becomes its own sub-set.
A
B
We may also find things that are in our universe, but
not in either of the sets.
A
B
A U B or A Union B is everything in either set
Mouse, dog, platypus, duck and goose
A ^ B or A Intersect B is everything in BOTH sets
In this case, just platypus
A
B
~ A or Not A or A Complement = Everything outside of A
Cow, duck, goose & snail
A-B or A minus B = Everything in A that’s not in B
Mouse and dog
A
B
~(A U B) or Not A Union B is everything outside of A and B
Cow and snail
~(A ^ B) or Not (A intersect B) is everything outside of the
overlap Cow, mouse, dog, duck, goose and snail