What is a Venn Diagram?
Any method in this world cannot replace the benefits of diagrammatic representations. The usefulness and ease of diagrams are so high that it does not require the end user to be literate, but any person from one corner can understand the diagrams describing any process or steps, from another corner of the world. Surprisingly, the origination of all the languages and the field of speech and expression itself got evolved on the basis of pictures, diagrams, etc.
These Venn diagrams speak for themselves and don’t need any specialization for the reader or the end user. Humans today have evolved in all spheres that even illiterate can decipher the topic or sense to be conveyed using diagrams. Also, pictures are the best way to retain the conveyed message. The magic that our visual memory does in terms of retaining objects, steps or processes is just ethereal.
Venn diagrams are one such diagram which could be put to use in almost every domain to depict the relation between two sets. The way it depicts, the requirements of making or drawing a Venn diagram and every such minuscule detail pertaining to Venn diagram can be learned while reading this article, exclusively on Venn diagram.
Venn diagram forms a part of set theory, wherein there is a set, and its contents are important. The changes or relations are forms amongst contents of two or more ‘sets.’ It has got definitions, symbols, diagrams, and its meaning. So it gives a clear picture of the exact connection amongst the contents the diagram is made for.
Venn diagrams are very useful while solving and getting to know about complex databases. For example, a conclusion has to be filtered out from random information under a title for a colony, Venn diagrams could be very helpful in churning out the exact and relevant information. This has its use in complex analysis of data, wherein a diagram or numerous Venn diagrams could be of sense in spite of going through the humungous sheets containing numbers only.
So to detail more on this useful tool, let’s understand the reason or historical background of the evolution of the Venn diagrams. Further, the symbols, signs, shapes used will also be discussed. And finally, if a tool could be of so help, then its uses must also be varied and concluding the topic will be in the last few lines.
Venn Diagram: Historical Background
The Venn diagrams have a long history. In earlier times, to get any discovery its proper shape and form as it may now take years of practice, research, and applications. Various scientists, logicians, mathematicians, and other such eminent personalities worked on for years to derive at a conclusion. And then there were cases of acceptance and recognition at the international forum too. Further, the category into which the discovery should be put was also a question.
Amidst all this scenario, the way Venn diagrams got recognized, tested and put into practice today is a combination of efforts by three eminent personalities. In the case of Venn diagrams, one mathematician carried forward the ideas and concepts of others, and finally, the Venn diagrams took shape.
So the final recognition could be given in a combination of those three mathematicians, who have given an indispensable discovery to people to be able to solve complex and chaotic problems.
Venn diagrams are basically a way to exhibit groups symbolically and ways to comprehend the links among the contents of these groups. It is a part of mathematics.
And there are various branches of mathematics, which require the prior knowledge of sets, on which Venn diagrams are based. So the base or background without which the Venn diagrams would be incomplete is sets.
And it was a German mathematician named George Cantor who came across the topic of sets while solving something else.
Talking about the depiction, it was the Swiss mathematician Euler who ideated a way to depict sets pictorially. He thought to present them through points in a closed curve.
Then, finally, it was the British mathematician and logician named John Venn who actually put things into place. The Venn diagrams are named after his name. He took further the idea of Euler and developed techniques to use the Venn diagrams for solving a lot of mathematical and logic-based problems and queries.
Requirements of Venn diagrams
Since Venn diagrams are a pictorial representation, there is a requirement of simple shapes, i.e. a rectangle and a circle. Each of them has its own meaning to convey. A rectangle depicts the universal group pertaining to the exclusive group about which we are talking. To make things simpler, consider a group which has in it all the words starting with the alphabet ‘s’.
If we are to sketch a Venn diagram using this group, its universal group or set will be all the words existing in the world with all the alphabets or even their combination. This universal group or set will be depicted using a simple rectangle. The dimensions are not fixed or are of any standard. It is as simple as making a normal rectangle on a sheet of paper. This group is depicted by an alphabet written in capital, i.e. ‘U.’
Next up is the important element of drawing a circle. Now, this is what makes a Venn diagram made in its entirety. So, finally what constitutes making a Venn diagram is disclosed, i.e. it has a rectangle in which a circle is also drawn. This is how a set or a group is made about which the whole point revolves is depicted.
Coming to the circle, it depicts the exclusive set or group whose contents will be linked, or this connection will be talked about. In the example cited above, the group containing all the words starting with the alphabet ‘s,’ will be depicted in a circle.
Inside the circle, if it is possible one can write the contents of the set in it. Again, the diameter or dimensions of the circles to be drawn are not any standard or fixed. This is a set and is depicted by any of the alphabets.
How to link and use the Venn diagram?
Now that we know how to sketch a Venn diagram consisting of a set, we get to a level higher and understand how to comprehend as to how to use and link sets. By sets, it is clear that, in a Venn diagram, there could be multiple sets. By this, it is in no way that one needs to draw multiple rectangles with circles in it. It is just one rectangle with many circles in it.
Now as per the data and contents of a set, it is likely that the circles may collide or overlap. It is this overlapping and non-overlapping area that is put to use, to get answers to mathematical problems.
One may get confused as to how and why would circles overlap or even won’t. Here is an example to make one aware of it. Consider two groups, one named s1 and the other being s2. The contents of the group s1 are all the novels written by authors whose name starts from the letter ‘a.’ while the group s2 contains the names of all the non-fiction novels.
Now it is a possibility that either the contents of the two groups have a common element or more than one, or another possibility exists wherein there will be no common content of the two groups. So, in either case, a Venn diagram is drawn, and there is a link formed.
By definition, the universal set for this example will be all the novels written in the world. Now if the two circles have one or more than one content in common, the two circles drawn inside the rectangle will be overlapping. Hence the common elements between the two sets will get displayed by the overlapping area.
And if the other possibility exists, then the two circles will be drawn with no area in them to be a common part. They won’t even touch each other. And a possibility of concentric circles does exist. So this is how they are linked.
Using the Venn diagram would always need some data in the form of numbers. So there has to be the total of the number of contents present in a particular set or group while making Venn diagrams. Further, numerical data regarding the problem is required to calculate and get to numerical conclusions. Here the total or the universal set’s data could be, or could not be required depending upon the nature of the problem.
Venn Diagram Symbols
To get to a logical conclusion, certain symbols are put to use. And after all the purpose is to make the complex, behemoth looking numbers or problems in a simpler manner. The symbols used are:
- Union: To denote all the contents of the sets involved but different from the universal set, the function or operation union is used. It is depicted by a broad U. It looks like the letter C is rotated 90 degrees to its left.
- Intersection: To denote the common or overlapping contents of the sets involved, we have this operation named as the intersection. It is symbolized by an inverted letter U.
- Difference: This shows the contents of a set not present in the other one. It has the same symbol of a Triangle/Delta – Δ.
These are the basic symbols required while working using sets and Venn diagrams. There are other than these too but usually aren’t used that frequently. It depends upon the nature of the problem and its answer. Further, these are used once between the names of the two sets and can be used multiple numbers of times to depict the desired connection between the elements of the sets involved.