As of last year, I began teaching a course in theoretical computer science, and one of the topics that we cover is sets. In particular, we go over:
- the actions $\cup$, $\cap$, and $-$,
- cardinalities
- Infinite sets in $\aleph_0$, and proving that sets are (or are not) in that set.
I thought that the last part would be tricky (particularly since we not only teach the diagonal graph proof, but also use the proof that hinges an element that points to the set of all elements that don't point to sets that contain themselves... I don't remember the name of that proof, but it is certainly cognitively tricky.)
I hit these last two subjects hard, doing quite a lot of practice and review. In the final examination, students did very well with the second two topics, but struggled mightily over (what I believed to be) simple Venn diagram questions like this:
Write an expression that represents the region in gray.
Some number of kids definitely got it, but for an uncomfortably large number of students, the answers they gave were almost comically convoluted, such as:
$((\mathbb{U} \cap (C \cup A)) - ((A - B) \cup (C - B))) \cup (A \cap B \cap C)$
... or were simply flat-out wrong. So, it's clear that I need to teach this better, and I will in the future.
My question, then, is thus: what are the cognitive traps that a teacher should be aware of when teaching union, intersection, and subtraction in sets?