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Venn Diagrams

2009-05-13 15:30:37 | 673 views | venn diagrams set theory euler elements members common

What this article is about and what it's not

This article will not go in deep on Venn diagrams, it will not give a detailed historical description on who John Venn was or his relationship to math.

It will however show the basics of Venn diagrams; how they are used, why they are used and what they actually represent.

Before reading more you should have a basic knowledge of Set theory and how it works.

What is a Venn diagram?

A Venn diagram is an illustration that shows the relationships between groups of objects by representing each set with, normally, circles. The circles overlap eachother to show all possible relationships between the sets. Here is an example:

A is a set that contains the values {1, 2, 3}
B is a set that contains the values {3, 4, 5}
C represents all values shared, in this case 3, since both A and B has that same value.

The C section is what's most important here. It shows the values the two sets has in common, in this case 3.


30 people gets interviewed and asked if they like McDonalds, Burger King or both. The result was that 17 people liked McDonalds, 8 people liked both McDonalds and Burger King, X people liked only Burger King and 7 people didn't like any of them.


What we got already is the following:

30 people participated
7 of the participants didn't like either one of them
8 liked both of them
9 liked only McDonalds ( 17 - 8 )
6 liked only Burger King ( 30 - 17 - 7 )
17 liked McDonalds ( 9 + 8 )
14 liked Burger king ( 8 + 6 )

Let's take another example:

Three people are playing a game where they are designated certain numbers between 1 and 12. If their numbers comes up they get a point. The numbers designated are as follows:

Player A: {1, 4, 6, 9, 11}
Player B: {2, 3, 4, 8, 10}
Player C: {4, 5, 8, 9, 11}

This is how we can depict it:


As we can see, there are two cases where nobody wins, 7 and 12. There is one number where everyone wins, 4. There are also two cases where two of the players win; 9 and 11 causes player A and C to win, and 8 causes B and C to win.
What we can see is that player B has the biggest chance to win individually since it holds 3 unique numbers where it stands as sole winner. C only has one number where it stands as the only winner.

If we pick 5 random numbers, 1, 4, 6, 8 and 11 the scoresheet looks like this:

Player A: 4 pts
Player B: 2 pts
Player C: 3 pts

Let's take a look at the overlapping areas. These can be read in different ways by using the ∩ and U notations. The ∩ symbol means "X and Y", the U symbol means "X or Y or both".
There is also the ' (apostrophe) to be used which means not.

Here is a picture that shows the relations between the players of the map:


Some examples from the picture:

A∩C = 4, 9, 11
(A and C, that means only elements common to both A and C)

AUC = 1, 4, 5, 6, 8, 9, 11
(A or C or both, that means any elements in either A or C)

(AUB)' = 5, 7, 12
(Not A or B, that means any elements that is not in A or B)

A∩B∩C = 4
(A and B and C, that means only elements common to both A, B and C)

Note that in the picture the area A∩C holds only 9 and 11, but really A∩C includes A∩B∩C, just like B∩C holds A∩B∩C since they are also common to the two sets.


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