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## Venn Diagrams

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**Venn Diagrams**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**B**A ~ 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