GMAT Integrated Reasoning: How to Interpret a Venn Diagram

In GMAT Integrated Reasoning, Graphics Interpretation questions will present you with a piece of data in the form of a graph, Venn diagram, scatter plot, etc. Below will be two statements, each with a missing portion. You will be asked to answer by choosing one of four choices presented in a drop-down menu.
So, what is a Venn diagram? And how can we apply our knowledge of set theory to the new Integrated Reasoning section? Let’s take a look at how Venn diagrams are currently tested on the GMAT first. This is a classic, challenging GMAT question involving a Venn:
 

We can quickly make sense of this word problem by creating our own Venn diagram:

Notice how the number at the top is the total for ALL PARTS of the Venn. The number above each circle “N/3” and “?” represent the total number WITHIN that circle (both the overlap and non-overlap sections). To solve, the key to understanding this question lies in the last sentence: who did not both receive a degree in the applied sciences and graduate from a school in one of six northeastern states?
We have two categories to sum: the people who ONLY received a science degree but NOT from one of the 6 schools, and the people who ONLY went to the 6 schools but did NOT receive a science degree. I made up variables for these categories (x and y).
If N = 12, there are 4 applied science students, 1 of which is both. That means x = 3. If 4 students are applied science, then 12-4 = 8 are from one of the six states but NOT applied science. y = 8. 3 + 8 = 1. Therefore, we are looking for an answer choice that gives us 11 when N = 12; the answer is A.
The key to a Venn is being able to apply information from one part of the diagram to another part.