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Visually Representing Numerical Data | Splitting up the Whole with Pie Graphs | Showing the overlap with Venn Diagrams | Comparing Categories with Bar Graphs | Seeking Trends with Line Graphs | Showing Orderly Data with Histograms | Functions as Graphs in the Coordinate System | Summary Showing the Overlap with Venn Diagrams Venn diagrams are another way to show how something is split up, but it is designed to work for the things being described can have more than one characteristic at a time. An easy example is a survey of a junior high school classroom to find out which students have a computer at home and which have a bicycle at home. In a classroom of 40 students, let us imagine that 15 say they have computers and 20 say they have bicycles. A single student can be characterized as "having a bicycle," as "having a computer," as "having both," or as "having neither." In other words, there are two categories of "having" and each student is classified as "yes" or "no" in each category. We need some visual way to show who has what. In a Venn diagram, we draw a "blob" to show all the students who have bicycles, and then we draw another "blob" to show all the students who have computers – but the interesting part is whether that second blob is completely separate from the first blob, overlapping some, or completely overlapping. Venn diagrams give a visual description of how many students are in the category of "having both." Let us first look at a Venn diagram designed to show that 5 students have both bicycles and computers. We will label each "blob" with a letter that is suggestive of what it measures, so "B" we will label having a bicycle, and "C" we will label having a computer.
Usually, we make it clear that we are only discussing the particular sample of 40 students by putting the "blobs" inside a box, and we also usually use shading or coloring to highlight the separate "blobs" (I’ll shade them blue and red so the overlap is purple). Thus the full Venn diagram looks like this:
One other thing we can do with the Venn diagram is put numbers in it to show how many students belong to each part. So we would put a 40 just outside the box to show it is representing 40 students. And we would put a 5 where the blobs overlap, because that’s how many have both computers and bicycles. And since 15 have computers, but we already put 5 in the overlap, we would put 10 in the remaining "crescent moon" shape. And similarly, 15 in the leftover bicycle crescent. And because the two blobs together are counting up 10 + 5 + 15 = 30, we would label the area outside the overlapping blobs as 10 to make up the total of 40 students. 40 The formal mathematical language for "blob" is set,
for "overlapping" is intersection, for the "whole
thing" is universal set and for measuring the "size
of" a set S, use the notation n(S). So for this example, n(U)
= 40,
n(B) = 20, n(C) = 15, n(A intersection B) = 10. The terminology used for
lumping things together is union, so n(A union B) = 30 because 30
students have either a bicycle or a computer or both. Finally, if
you don’t want to use the words, the symbol for intersection is
The best part of a Venn diagram is you can modify the picture in ways that visually tell the story of what is different. For example, look at the picture below and make up a sentence about owning bicycles and computers. 40 Wasn’t your sentence something like, "All students with computers also have bicycles" or something of the sort, such as, "Ten students have both bikes and computers, but five more have only bicycles"? The mathematical terminology for one set being inside the other is to call the one inside a subset of the other one, so in this case we would say, "C is a subset of B." The other extreme, that of no overlap at all, can also be visually depicted: 40
Now let's do Practice Exercise 17-1 Please go to the links below for additional instruction for this topic included on the HSEE: Venn Diagrams:
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and for union is U. If you study sets in
your further mathematics, you’ll find out an interesting
equation that relates the sizes of the sets to the sizes of the
union and intersection. Can you guess that equation by looking at
the diagram and thinking about this example?
