# Sets, Matrices and Venn Diagrams:

by on December 21st, 2009

Sets, matrices and Venn Diagram – they are all the same. It’s really a matter of preference; some students like to use Venn diagrams, and others make matrices. Personally, I prefer a matrix format, but there is no “better” way.

On your GMAT, you will encounter 1-3 questions that contain overlapping groups with specific characteristics. You will almost never see more than two characteristics (since you can’t draw 3D on your scratch paper). For illustration, let’s take a look at the following Data Sufficiency example:

Of the 70 children who visited a certain doctor last week, how many had neither a cold nor a cough?

(1) 40 of the 70 children had a cold but not a cough.
(2) 20 of the 70 children had both a cold and a cough.

There are two characteristics (cough and cold) and two categories for each (yes and no), so there are four total categories, as indicated by this matrix:

I’ve filled in the given information from both statements, and the parenthetical information is inferred. This clearly lays out the 4 combinations of options. If we sum vertically, we can infer that there are 60 total children with colds. Because there are 70 total children, this also means that 10 do NOT have colds. The bottom-right quadrant cannot be found because we do not know how those 10 children get divided between the two empty boxes. Choice E – together the statements are insufficient.

We may also visualize the question as Venn diagram, in which there are still two characteristics, represented by overlapping circles. You will notice that there are still two undefined regions, so the given information is insufficient.

For any Data Sufficiency or Problem Solving Set question, map out the provided information and mark the region that you need to find. Note that there may be implicitly defined regions, such as “60 children have a cold” above. Let’s look at one more example in matrix format.

Each of the dogs in a certain kennel is a single color.  Each of the dogs in the kennel either has long fur or does not.  Of the 45 dogs in the kennel, 26 have long fur, 17 are brown, and 8 are neither long-furred nor brown. How many long-furred dogs are brown?

A. 26
B. 19
C. 11
D. 8
E. 6

Again, we are provided a small amount of overlapping information and our matrix can simplify our visualization. The parenthetical information is inferred. In fact, in order to have found the correct answer (6) in the top-left quadrant, we had to have derived either of the two empty quadrants.

Of all the types of Quantitative questions on the GMAT, overlapping sets are some of the most like puzzles. This should make you very, very excited. Because who doesn’t like puzzles?