increasing the set capacity of Venn diagrams with color codes

[MonDie]

I should point out that the term "set capacity" is totally made up and informal.

As always, you can alter and/or repost my images, and you don't need to give credit because they aren't copyrighted.

 

I learned about Venn diagrams recently. When I attempted to make a 4-set diagram, I realized that the classic intersecting circle approach didn't cut it. Where A and C are opposite circles in the diagram below, A and C don't have an intersecting region that doesn't also intersect B or D.

 

 

According to Wikipedia, this is actually a Euler diagram.

http://en.wikipedia.org/wiki/Venn_diagram#Extensions_to_higher_numbers_of_sets

 

Wikipedia gives some alternatives for drawing 4, 5, or 6 set Venn diagrams. However, those diagrams are so confusing that, instead, I added a color dimension to increase the capacity of the simpler, 3-set diagram. The image below (Figures 1 to 3) shows the process of converting a spacially-denoted set into a color-denoted set. Where an element is a dot placed somewhere in Figure 1, the element's memberships are denoted solely by its position on the diagram. Figure 2 loses the upper region, but it still gives just as much information. The upper region was actually exchanged for a color scheme in which elements belonging to the set are blue. Now, an element's memberships are denoted by its position and color. In Figure 3, the various colors present in a region are combined into a overall region color (or shading). Regions with blue or red elements become blue or red, and regions with both become purple, the color obtained when you mix blue and red paints.

 

 

Although I was doubtful about performing operations with this scheme, I think it will be quite easy. For example, imagine that A and B are spacially-denoted sets, and membership in C is denoted by blue color.

A ∩ C would simply mean: disregard all elements except the blue elements in A.

A ∪ C would be read as: disregard all elements that are neither blue nor within A.

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Things get more compicated when you want a color scheme that denotes 1< sets. I have given some suggestions in the image below.

Both pictures outline color schemes based on the primary colors of light.

The first scheme (top) increases the set capacity by 2. The single set regions are red and blue, and the intersection is green. When these colors are combined to determine the shading of a spacial region, they mix easily (red + green = yellow; red + blue = violet; green + blue = cyan). The elements that fall outside all color-denoted sets are gray, and they might effect a region's shading by lightening or darkening it.

The second scheme (bottom) is a color scheme for three sets. Right now, you're thinking it will be simpler than the prior scheme, but you're WRONG. When you mix these variously colored elements to determine a region's shading, it's difficult to avoid redundancy. For example, white might denote that there are only white elements, or white might denote the presence of all primary colors. Also, how do you denote the presence of both black and white elements? Do you dim the color toward gray?

Of course, you could bypass this problem altogether by leaving the colors separate, but this may be harder to remember, and it may not look so pretty.

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Feel free to introduce new terms or phrases. If similar ideas have been outlined elsewhere, I would like to see how others did it.

Edited June 7, 2013 by Mondays Assignment: Die

[John]

So now we'll have Venn diagrams, Euler diagrams, and Mondays Assignment: Diagrams.

[MonDie]

So now we'll have Venn diagrams, Euler diagrams, and Mondays Assignment: Diagrams.

 

Although I didn't know it when I came up with my username, the "Die" is actually the singular of dice, not the verb to die.

Edited June 7, 2013 by Mondays Assignment: Die

[John]

Not to get too far off-topic, but my discrete mathematics professor annoyed me throughout the semester I spent in her class by referring to a single die as "a dice." If I had a soul, that destroyed it.

[MonDie]

In highschool, my psychology teacher got angry and said I have selective hearing.

 

I misread your post.

Edited June 7, 2013 by Mondays Assignment: Die

[Bignose]

It is a nice idea, just not universal. By which I mean it doesn't help people who are color blind, for example.

[MonDie]

A color-blind person could still read the basic single-set scheme. The single-set scheme would be useful in instances where a single set contains roughly half of the elements.

Edited June 7, 2013 by Mondays Assignment: Die