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Their history began in 1881 with the publication of a book by John Venn, entitled Symbolic Logic. This introduced various diagrams which were used to illustrate complex logical relationships and give them an intuitive appeal for students who had not yet developed facility in symbolic logic. Among these was what is today known as the Venn diagram.
In his book, Venn described ‘a new kind of diagram’ which was the forerunner to today’s diagrams and added: “The use of these curves is in no way limited except by the laws imposed upon good taste.” He later wrote that he had derived them as far back as 1877 when he was asked to contribute an article on mathematical topics for a book in memory of the English mathematician Augustus de Morgan. The diagrams then became known as Venn’s circles and were initially viewed with scepticism by some mathematicians. It wasn’t until 1883 that his friend, Francis Maitland Balfour published a paper on them under the title: The Principles of Logic as Based on the Theory of Classes. He said that Venn’s diagrams enabled a “new and more general” method for dealing with logical problems, including some which had previously appeared to be insoluble.
Venn’s work in symbolic logic had a profound influence on mathematics and philosophy, especially the fields of Boolean algebra and predicate calculus. Many other academics have since expanded upon his original ideas to create new concepts with Venn diagrams including Henry Dudeney who studied their use as puzzles while Martin Gardner popularised their use in their current form.
Basic Venn Diagram: The simplest type of diagram consists of two circles which represent the sets being studied. Inside each circle, any points where they overlap indicate that there is an item in both sets while areas outside them are excluded from either set.
Intersection (or “overlap”) of sets: The set which is the intersection of these two circles consists of those items that are in both A and B.
Union (or “sum”) of sets: This shows all possible combinations between the members within each circle, including any duplicates to show multiple possibilities such as AB or BA . It also shows that the C set is empty because there are no members in both A and B.
Combinations of sets: This shows all possible combinations between the members within each circle, including any duplicates to show multiple possibilities
Set Theory is a branch of mathematics which deals with the study and classification of all possible objects, including how they interact. It differs from ordinary arithmetic in that it does not deal directly with numbers but rather uses sets to represent them as well as other types such as logic values (true or false) and shapes like squares and circles.
Set theory has been used for many purposes including: logic, mathematics, science (especially biology), philosophy, linguistics and computer programming. By working with sets instead of numbers it is possible to analyse mathematical statements at a higher level than their ordinary arithmetic counterparts can achieve because the set’s members are not restricted to numbers.
For example, if you want to consider the set of all even prime numbers it would be impossible in arithmetic because they contain no natural number members while a Venn diagram shows that there is exactly one such subset (2 & 5) and therefore can also show equivalences between them such as:
The sets of all even prime numbers and odd non-primes are equivalent because they contain the same members. The set of primes is not a subset (or part) of the set containing only their powers, except for two cases where it actually contains them in its power elements! Since these are both even numbers, the set of all prime powers is equivalent to that containing only its odd elements.
While arithmetic can tell you if a statement like “2 + 3 = 5” is valid or not it cannot express why this might be so and therefore doesn’t provide any insight into how sets interact with each other. In this way, set theory can be used to examine the logical relationships between sets and identify which ones are equivalent or even contradictory (incompatible) using a Venn diagram rather than more complex logic expressions.
Mathematics uses Venn diagrams in several different ways and they are particularly useful when dealing with sets of numbers, shapes or logic values. They also often appear as puzzles to illustrate their logical relationships but can be used for more than this purpose too such as exploring the differences between topological spaces (spaces which can be continuously transformed into one another).
In fact, even if you are not familiar with Venn diagrams in mathematics and logic they may still be present within your daily life. You probably use them all the time without realising it! For example; on any piece of packaging for food products there is a diagram showing which types of foods should or shouldn’t be stored next to one another. Another example could be in a marketing strategy where the Venn diagram is used as an aid for product placement, i.e: placing products together on shelves which will appeal more to customers and thus increase sales potential (even though they’re not technically mathematical examples!).
Venn diagrams aren’t to be confused with Euler diagrams, which are more similar to a Venn diagram. The main difference between the two is that Euler diagrams combine multiple sets, whereas Venn diagrams represent relationships like “A or B” and “not A.”
Euler diagrams are named after Leonhard Euler, who developed them in the early 18th century to illustrate complex mathematical concepts. He was actually looking for a way of solving problems with his students that didn’t involve pen and paper at all (a very different approach from our modern world of Mathematica).
Euler diagrams have a wide range of uses. A famous example is the Eulerian path: connecting all city names on an itinerary to form one continuous trip in order, minimising crossings over water or land when necessary. Another application involves creating Facebook posts that mention different topics (politics and art) while keeping the total number of words at a minimum.
Venn diagrams and Euler diagrams are similar in nature, but it’s essential to know the difference between them if you want your research paper or presentation slide deck to be free from errors.
