This week on Occupy Math, we look at Venn diagrams. Most students encounter these early in their education and they seem pretty simple. In this post we not only look at the math, but humor and the use of Venn diagrams as a creative prop. We will also see that there are still graduate research level questions about Venn diagrams. A Venn diagram takes different sets of objects and diagrams them as circles — or other shapes — with objects in both sets appearing in the intersections of the circles. Optionally, the area of different parts of the diagram can give you information about the size of the set of things in that part of the diagram. The example at the top of the post is about the “positive whole numbers”. This is called the universal set, which is fancy talk for everything the diagram covers. The special sets shown in the diagram are the even numbers and the prime numbers. Since all the primes except 2 are odd, the only thing in the intersection of the even and prime numbers is 2, which the diagram shows.
The Venn diagram on science fiction monsters, below, is partly filled in. There are other types of science fiction monsters, but a Venn diagram can focus on just some sets. Mad scientists, for example, are not marked and you can think of them as being in the white area outside the three circles. This diagram focuses on zombies, robots, and aliens, but leaves blank the intersections of things in common with zombies and robots, with robots and aliens, and with zombies and aliens.
Now let’s include all the two-way intersections. Zombies and robots are things that do not have emotion. Zombies and aliens both have a taste for human flesh. Robots and aliens both have access to high technology. In this case the intersections are showing common qualities that the two types of things have. Note that the triple intersection is still blank — what do zombies, robots and aliens have in common?
Since this Venn diagram is about bad science fiction, the quality that all three groups have in common is that they want to kill all humans! This Venn diagram is intended to be at least a little funny. If you perform an internet search on “Venn diagram jokes”, this one will probably show up along with many others. This example also gives us a sense of how Venn diagrams can be used as a creative prop. Try giving students the top diagram — with only the three categories filled in — and ask them to fill in the other four areas. Since pop culture supplies the “rules” for this situation, this is an exercise in creativity, not precision mathematics.
Venn diagrams can be dangerous!
An instructor might use the two-category Venn diagram below to explain to students how important effective studying is. There are two categories, but the circles do not overlap, and so there are no common elements in the two sets. Suppose that a student who has seen this Venn diagram has flunked a test because their dog just died and they are upset. This can make the apparent accusation “you did not study effectively” seem harsh.
This is the danger: humorous or cautionary Venn diagrams are pretty simple and it is easy to place an interpretation on them that the author did not intend. A student who flunks a test because they are upset about their dog’s death probably really did not study effectively. The thing is, this is not an accusation, it is simply something that happened — it may be true that you failed to study effectively, but that may not have been your fault. Another low-fault reason that you might not study effectively is that you had four examinations the same week. Time is finite. Maybe you could have started studying earlier, but it might not have been practical. Also, if your dog dies, talk it over with the instructor. They might have a way to give you a delayed exam.
The meaning of intersections
One way to avoid controversy in your Venn diagrams is to stick with just math. The Venn diagram below has three categories: odd numbers, even numbers, and multiples of three. Since there are no numbers that are both odd and even, those two sets are shown as not intersecting.
The two intersections shown are the odd multiples of three and the even multiples of three. Rather than put in those words down on the diagram, the intersections are shown by giving several examples. This is acceptable if there are enough example to get the pattern. One might also say 6n and 6n+3 where n is a whole number.
Using a Venn diagram to solve a math problem
Suppose we are supposed to answer the question, “how many numbers in the range one to one hundred are not multiples of two, three, or five?” We could write out the numbers from one to one hundred, cross off the ones we do not want, and then count the remaining ones, but that would be a lot of work. Half the numbers are even, one third (rounding down) are multiples of three, and one fifth are multiples of five. In the range one to one hundred we have 50 even numbers, 33 multiples of three, and 20 multiples of five. Totaling up these undesirables we get 50+33+20=103. Since there are only 100 numbers, it is clear that the problem cannot be solved by counting the categories of undesirables, adding them up, and subtracting them from 100. Let’s make a Venn diagram to figure out why this will not work.
We start with the three categories, the multiples of two, three, and five. When we fill in the intersections, multiples of two and three are multiples of six, for example, then we see the problem. The 50 even numbers included 16 multiples of three, so those got counted twice. Using the same technique (divide and round down if you get a fraction) we see that there are 16 multiples of six, 10 multiples of ten, and 6 multiples of fifteen. Each of those was counted twice and we need to subtract them out of the total. Except that when we do that, we have another version of the same over-counting problem. When we subtract the multiples of six, ten, and fifteen, then we subtracted the multiples of thirty out three times. We had already included them three times (once in each of the original three categories) so we need to add them back in once to let the multiples of thirty be counted once, each. There are 3 multiples of thirty in the range one to one hundred, by the divide-and-round-down method.
This means the number or numbers are divisible by two, three or five is: 50+33+20-16-10-6+3=74. Since we have 100 numbers there are 100-74=26 that are not multiples of two, three, or five. Check this the long way to make sure it is right, if you have doubts. Notice that the Venn diagram steers our problem solution for us by showing us all the relevant categories.
Occupy Math chose one to one hundred as the universal set for this problem because it is possible to check on paper by crossing off the numbers you do not want. Occupy Math actually did this as a check on his addition and subtraction. Suppose we asked about the numbers from one to one thousand, instead of just going up to one hundred. Then the numbers that are multiples of two, three, or five, etc. would go like this: 500+333+200-166-100-66+33=734 so there are 1000-734=266 numbers from one to one thousand that are not divisible by two, three, or five. Doing that by hand would be quite a chore — but the method works for any size number where you can divide and round down.
Venn diagrams can be used to diagram problem spaces, you can have students fill them in to demonstrate that they understand the relationship between mathematical categories, or you can use them for creative exercises in which you pick a few categories and have the students try to fill in the intersections. If you click the link at the top of the post, to an article on Venn diagrams, it will show you the issue of making Venn diagrams for more than three categories. It is tricky. Below are blank Venn diagrams, including one that uses ovals to make four categories. Occupy Math includes them so you can make your own Venn diagrams. Finding Venn diagrams with many categories is an active area of mathematical research.
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics