There are many ways to look at groups of items and tell what is alike and different, but you'll always need some kind of system for organizing what you see. This is especially true if you want to make or test a general statement about the items and the groups they form. Let's look at how organizing items can help you draw conclusions.

Out of forty students, 14 are taking Geometry and 29 are taking Biology.

• If five students are in both classes, how many students are in neither class?
• How many are in either class?

In order to answer these questions, we can use a Venn Diagram to help us organize the students into groups. Follow along in the slide show below to see how to do this.

To begin, draw two overlapping circles labelled with the total for each of the two classes. Since we know that five students are taking both classes, enter "5" in the overlapped portion. Since there are 14 Geometry students, and 5 of those students are in the overlap, we are left with 9 students. So nine students are taking Geometry but not Biology, and we can place 9 in the "Geometry only" part of the "Geometry" circle that is shown on the left. Similarly, since there are 29 Biology students, and 5 of those students are in the overlap, we are left with 24 students taking Biology but not Geometry. We can place "24" in the "Biology only" part of the "Biology" circle shown on the right. Finally, if you add 9 + 5 + 24 = 38, you can see that 38 students are in either Geometry or Biology (or both). This leaves 2 students nowhere in the circles, so they must be the students taking neither class.

Generally, diagrams help you figure out relationships among items in different groups -- especially when some items belong in more than one category. In this example, we had several categories -- Geometry, Biology, both, and none. A Venn diagram like the one above is the approach used most often in geometry to show these kinds of relationships.