Venn Diagrams

What is a Venn Diagram?  The Venn diagram was introduced by John Venn.  It is a diagram that shows all possible logical relations between a finite collection of sets.  A Venn diagram consists of overlapping circles. Each circle contains all the elements of a set. Where the circles overlap shows the elements that the sets have in common. Generally there are two or three circles.  Sounds a little complicated and can look that way too!

A 5 set Venn diagram

The intersection of the Greek, Latin and Russian alphabet.

Lets start with a couple definitions and some examples to see if we can make this less complicated.
A set is a collection of objects.
A ∩ B is the intersection of two sets or all items common to A and B.

How are Venn diagrams useful in math? Venn diagrams enable students to organize information visually so they are able to see the relationships between sets of items.

 
Venn diagrams can be used for to compare other sets too.  Lets discuss its use in the life of a College Student.
Let A = enough sleep, B = social life and C = good grades.
Then,
A ∩ B = Slacker
B ∩ C = Zombie
A ∩ C = Nerd
and A ∩ B ∩ C = Defies the Laws of Physics

This can be shown in the following Venn Diagram.


















Lets try another example involving food.

If A = Flour, B = Egg and C = Milk
Then, 
A ∩ B = Pasta
A ∩ C = Batter
B ∩ C = Omelette
and
A ∩ B ∩ C = Pancakes

Problem Solving Strategies

Math.  A little word that strikes fear among so many students.  A jumble of numbers and letters in a problem and you don’t know how to solve it.  Here are some strategies you can use when trying to solve a math problem.

1. Guess and check

2. Make an orderly list

3. Draw a diagram

4.    Look for a pattern

5.    Make a table

6.    Consider special cases

7.    Use a variable/Use two variables

8.    Work backwards

9.    Eliminate possibilities

10. The pigeonhole principle

11. Inductive reasoning

12. Deductive reasoning

Most strategies listed seem pretty self-explanatory.  But what is the pigeonhole principle?  I don’t remember learning about that in school.

By definition, the pigeonhole principle state, if m pigeons are placed into n pigeonholes and m > n, then there must be at least two pigeons in one pigeonhole. 

Example 1

If we have 10 pigeons and 9 pigeon holes, then m=10 and n=9.  Therefore 10 > 9 so there must be at least two pigeons in one pigeonhole.

Example 2

If we have 26 pigeons and 25 holes, then m =26 and n=25.  Therefore, 26 > 25 so again there must be at least two pigeons in one pigeonhole.

Example 3

If we have 7 pigeons and 9 pigeonholes, then m=7 and n=9.  Therefore, 7 < 9 and the pigeons can each have their own hole. 

See, math isn’t so hard!  The pigeonhole principle seems pretty simple to understand.  If you don’t have enough space somebody has to share.  Peter Gustav Lejeune Dirichlet is believed to be the first to formulate this idea.  Click on the links to learn more about Peter Gustav Lejeune Dirichelt and the pigeonhole principle.