You may notice that there is some connection between this and this example. In the first we have the intersection graph of all attendance lists of the courses, whereas in the second we are considering the intersection graph of all course lists of the students. Both sets of lists obviously contain the same information. Each one is the dual of the other.
Duality is the most important, yet very natural, notion for intersection graphs.
Look at an example. This essentially formal definition gets life by looking on intersection graphs. We may think of the a A as detectors reporting all hyperedges covering it. This is equivalent to the two statements above. Look at our example on how the star graphs of an intersection graph may look like.