Venn diagrams and Matrices

i received a private message regarding this thread.

as in the case of many other strategy-related issues, this one contains a large component of individual preference. if you are overwhelmingly familiar with one or the other of these devices, then you'll probably find that device “easier” in any case to which it's actually applicable.

however:

* if a problem contains three overlapping sets, then the “matrix” (which, by the way, is more commonly known as a two-way table, in statistical parlance) is actually impossible to make. (if there are three sets, then the “matrix” would actually take the form of a cube, which can't reasonably be drawn on a scratch pad.)
therefore, in three-set problems -- rare as they may be -- you're going to have to use a venn diagram.

* in two-set problems, you can usually use either of the devices. however, in most situations, the matrix/two-way table will involve significantly less work, because (a) it has more spaces and so can represent more quantities explicitly, and (b) its rows and columns are set up for automatic addition/subtraction.
for instance, consider a case in which students are either juniors or seniors, male or female. let's say you are given the following humble statement: “50 of the students are seniors.”
- in the two-way matrix, the statement is no problem: just find the intersection of “seniors”/“total” and write in 50.
- in the venn diagram, this statement isn't nearly as easy to deal with, because there is no single region that corresponds to the total number of seniors. there may be a circle in the diagram that corresponds to this quantity, but that circular region will be divided into two parts by the other circle. therefore, you will face a choice between two options, each of which is rather undesirable: either (a) make up a variable “x” (which you didn't need in the matrix table) and split the region into x and 50 - x, or (b) indicate on your diagram, with inevitable awkwardness, a single quantity that corresponds to the sum of the entire circle.

my short answer, therefore, is this: venn diagrams are required for three-set problems, but the two-way matrix is generally a better choice for two-set problems. but, again, individual comfort levels with these devices also play a large role.