 Let us look now at making a visual representation. In a graduating class of 50 students, 28 took calculus, 26 took physics, and 20 took both courses. How many students in the class did not take either calculus or physics? This problem is very easy to solve if we use a Venn diagram. It can be solved in other ways, but this is by far probably the easiest. Venn diagrams are representations of sets, that is one set. This is a second set. Let us say this is the set of students who took calculus, and this is the set of students who took physics. We can also represent the universe here. This is the total number of students. We know that there are 50 students in our universe, and let us write here the number of students who took neither calculus nor physics. We know that there are 20 students who took both courses, and that there are 28 students who took calculus. I write this outside of the circle, not inside, because I already have 20. I only need 8 more to complete the 28 students. Likewise for physics, I already have 20 who took physics, so I need 6 more for total 26. Now, 20 plus 8 and 6 is 34. I know that 34 students took either one of those courses, or both of those courses, and I have a total of 50 students, so I have 16 students who took neither physics nor calculus.