# Solving Problems Using Venn Diagrams

But since they also serve to make the problem look more confusing, we can often reduce the names to abbreviations, D, E, and S to keep things simple. One is purely algebraic: we write a series of equations based on the given information and try to solve that set of equations. I usually find the visual approach easier to use, but sometimes for a complex problem I resort to an algebraic approach. It’s fun making up nonsense names; and they emphasize the abstract nature of such a problem — it could be about anything! Doctor Greenie took it: There are two basic approaches to a problem like this.

In case you didn’t notice, the names are just Fred, Jane, and Davis, backward!

The "E" circle represents all the Enajs; the "D" circle represents all the Derfs; and the "S" circle represents all the Sivads.

And there is a special way of saying "everything that is not", and it is called "complement".

Last time we looked at various 2- and 3-set Venn diagram problems (and alternative methods). So let's set up both the algebraic and visual approaches to this problem and see how we can use both of them to solve your particular problem.

For example, the items you wear is a set: these include hat, shirt, jacket, pants, and so on.

You write sets inside curly brackets like this: You can also have sets of numbers: Each friend is an "element" (or "member") of the set. Now let's say that alex, casey, drew and hunter play Soccer: Soccer = (It says the Set "Soccer" is made up of the elements alex, casey, drew and hunter.) And casey, drew and jade play Tennis: Tennis = We can put their names in two separate circles: You can now list your friends that play Soccer OR Tennis.This information tells us that the combined number of elements in regions d and g (the elements that are both Enajs and Derfs) is one-third the combined number of regions a, d, e, and g (the total number of Enajs).Algebraically, we have d g = (a d e g)/3 This piece of information is equivalent to saying that there are twice as many elements in regions a and e together as there are in regions d and g together; so the above algebraic equation is equivalent to the following: a e = 2(d g)(3) Half of all Sivads are Enajs Again, we can't do a lot with this information just yet in our Venn diagram; this information tells us that the combined number of elements in regions e and g (the elements that are both Enajs and Sivads) is one-half the combined number of regions c, e, f, and g (the total number of Sivads).Item (6) tells me that the total number of Enajs is 90; item (2) tells me that there are twice as many Enajs that are NOT Derfs as there are Enajs that are Derfs.From this I conclude (10) a e = 60 and d g = 30 Then I finish the problem by combining (10) with (8) to get (11) a = 53 and by combining (10) with (7) to get (12) d = 29When I'm all done, as a check of the work I have done, I label each region of my Venn diagram with the numbers of elements I have determined for each region and go back and verify that those numbers fit all the given information.One discussion I found while collecting them deserved to be set aside for special examination, if only because it would scare the beginner. Here it is, from 2003: Derfs and Enajs: Algebra and Venn Diagrams All Derfs are Enajs. For the visual approach, we use a Venn diagram, consisting of three mutually intersecting circles that represent the Enajs, Derfs, and Sivads. And sometimes using a combination of the two helps to keep track of what I am doing towards solving the problem.This is called a "Union" of sets and has the special symbol ∪: Soccer ∪ Tennis = Not everyone is in that set ...only your friends that play Soccer or Tennis (or both).The one that I think is easiest is apparent (to me) with the Venn diagram but rather difficult to see with the algebraic approach.So let me make my explanation using the Venn diagram instead of the algebraic approach.

## Comments Solving Problems Using Venn Diagrams

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