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Lesson 154: Working with Venn Diagrams
We use Venn diagrams to help us solve problems involving sets that share some elements in common. It is an easy topic once you know how it works.
Here is a typical problem involving a Venn diagram: Out of 50 students, 18 are studying art, 27 are studying music, 4 are studying both art and music. How many students are studying neither art nor music?
The
easiest way to solve this problem and avoid error is to make a
diagram of the information. We can use what is called a
Venn diagram. See the picture at left.
You can see that the total number of students has been labeled in
the diagram. One circle shows the number of students taking
art, including the students taking both subjects, and the other
circle shows the number taking music, again, including the ones
taking both. It also shows that 4 students are taking both
music and art.
To figure out how many students are taking neither music nor art, first we must find exactly how many students are taking just art, and how many are taking just music. There are 18 students taking art, but 4 of those are also taking music. That means there are 18-4=14 students taking just art. Similarly, there are 23 students taking just music. There are 4 students taking both. That means we have 14 + 23 + 4 = 41 students accounted for. It's easy to now see that there are 50-41 = 9 students who are taking neither music nor art.
Note that there is another way to solve this problem, but you just need to be careful to not mix up the two methods. We can also add 18 for the art students, plus 27 for the music students, giving us 45. Now we have to subtract 4, because the students taking both classes were counted twice--once as part of the art group, and once for part of the music group. That will get us back down to 41 students, which is correct.
There are many variants of problems like this, but they can all be solved in this manner. Draw a diagram like the one above, and fill in all the data. Be very careful to not count elements more than once when you're doing your arithmetic. Also be careful to remember how you are using your circles. In this case, the values in my circles for music and art included the students who were taking both. You can also label the circles with values that exclude the students who are taking both, as long as you remember which of the two methods for labeling you used, and do your computations accordingly.
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