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When to Venn

Many students rely exclusively on the double set matrix when solving overlapping sets problems, but there are many situations where the double set matrix is overkill and a Venn diagram is a more efficient way to solve the problem.

I know some of you will be unconvinced, but let me make two points before the prophets of the double set matrix accuse me of heresy.

  • Venn diagrams are the only reasonable way to solve problems with three overlapping sets.
  • Venn diagrams are not only useful in overlapping sets problems. You can use them to solve probability problems, and they are also useful when finding the GCF (greatest common factor) or LCM (least common multiple) of two integers

So when should you use a Venn diagram, and when should you use a double set matrix?

A double set matrix will be most useful when you have a single set that can be split into mutually exclusive groups in two different ways.

For example, in a group of people we can split them up by looking at man vs. woman and red hair vs. non-red hair. We could also do man vs. woman and red hair vs. blond hair vs. brunet hair vs. bald.

  Men Women  
Red Hair Total Red Hair
Red Hair Total Red Hair
  Total Men Total Women Grand Total

Venn diagrams are more appropriate when there are two or three sets that share some elements. For example, a set of high school students, some of whom are in band, some in orchestra, and some in both (don’t forget that some students may be in neither band nor orchestra). We can also do three sets (or any number of sets for that matter). For example, kids in the chess club, kids in the science club, kids in the math club, and kids in two or all three of these clubs.

To summarize:

  • Use the double set matrix is you have one set of elements that can be split into two distinct sets in two different ways: Men and Women vs Red Hair and Not Red Hair
  • Use Venn diagrams when you have more than one set, and these sets share some elements: Kids in Band, Kids in Orchestra, and Kids in both Band and Orchestra