John Venn had an awesome beard.
His legacy, however, is a pictorial representation of overlapping sets that was named after him: the Venn diagram.
Look at that beard one more time.
Go ahead, I’ll wait.
Now if a guy has a beard that good and is remembered for something other than his beard, that other thing must be pretty special.
A Venn diagram is simply two overlapping circles, however, those overlapping circles create four separate regions. The red region represents the items that are part of Group 1, but not part of Group 2. The blue region represents the items that are part of Group 2, but not part of Group 1. The purple region represents the items that are members of both sets, and the white region represents the items that are members of neither set.
In equation form, this is written as follows:
Total = G1 + G2 – Both + Neither
However, in my experience, the graphical version of this analysis is much more helpful in allowing you the pick out the piece of information that is needed without any unnecessary confusion. The best strategy when you see overlapping sets is to begin by drawing a Venn diagram and filing in the known information. Consider the following example:
Ninety-two sports agents gathered for a conference on trends in free agency. Fifty-six of the agents had been in the industry more than ten years, but did not represent players in multiple sports. The number of agents who represented players in multiple sports was twice as many as the number of single sport agents who had been in the industry less than ten years. What is the maximum possible number of agents who represent players in multiple sports and have been in the industry more than ten years?
Trying to drop this paragraph of information neatly into a formula is asking for trouble, however, if your first reaction is to see two overlapping sets and draw a Venn diagram, you’re on the right track.
The next step is to add in the information that the question provided.
Make note of the fact that the 2X covering the multi-sport agents encompasses that entire circle, including the overlap. Using this graph you can now clearly see that the three regions we have values for cover all 92 conference attendees, so 56+2X+X=92. Solving this, we get that X=12.
To finish our problem, in order to get the greatest possible number of agents into that central overlapping region, we’d simply take all 24 of our multi-sport agents and put them in the middle, yielding our answer of 24.
When you face overlapping sets, remember to use a Venn diagram to represent your information because it’s an incredibly useful tool, and not just because he had an awesome beard.