Part of my schedule this term is an Intro to Set Theory. Just once a week, lunch time for the kids. I have a nice paper, An Introduction to Elementary Set Theory (Mathematics Association of America). We are reading the document, every line, and doing every exercise. It is reading/seminar style.

I like that the authors spend time connecting Set Theory to its history, and to Cantor and Dedekind. I added in some additional historical background – what was happening in math in the 19th century. I’d like to read more of the history. I get the feeling that the reexamination of Euclid and the development of set theory and the axiomatization of arithmetic and the development of logic are all part of a movement – but I don’t really know that, and I would like to learn more.

It is a great opportunity to introduce notation, to dwell on correct language. It is also their first heavy dose of proof by contradiction (indirect proof). And that is useful.

Today we were looking at subsets, and of 6 exercises, 4 were best answered with a proof by contradiction. A kid asked, a bit sadly, if they were ever going to be allowed to do direct proofs again.

It’s just 10 kids. At least 6 of them are there so that they can study Arithmetic with me in the spring (“Axiomatic Arithmetic”) – which I will describe some other time – but I insist that they do at least one proof-based course first. In the spring there will be a heavy dose of induction. I can imagine a kid asking, sadly, if they will ever be allowed to do proof by contradiction again…

In any case, today we were playing with subsets, and proving some basics. $A \subseteq A$ but $A\not\subset A$. And we talked a little about the empty set. And we had the annoying discussion about the empty set being a subset of another set.

Here’s the talk. One set is a subset of a second set if everything in the first set is also in the second. But we like stating this backwards. The first set is NOT a subset of the second if there is something in the first that is not in the second.

• A = {p, q, r, s}
• B = {q, r, s}
• C = {q, r, s}
• D = {s, t}
• E = {}

OK, so B is a subset of A. Everything in B is also in A. Or, there is nothing in B that is not in A.

B is a subset of C. Everything in B is also in C. Or there is nothing in B that is not in C.

D is NOT a subset of A. Why? Because t is in D, but not in A. (see how that works?)

Now, is E a subset of A? Is everything in E also in A? Hmm, that might cause an argument. Let’s look at it the other way. Is there something in E that is not in A? Nope. Then E is a subset of A

One part of this that’s fun and annoying is that while the concepts can seem slippery, we are doing them with all new notation.

t is in D? $t \in D$

t is not in A? $t \notin A$

B is a subset of C? $B \subseteq C$

A is a subset of B is equivalent to saying that everything in A is also in B? $A \subseteq B \iff \forall _{x}, x \in A \rightarrow x \in B$

A is not a subset of B is equivalent to saying that there is something in A that’s not in B? $A \nsubseteq B \iff \exists _{x}, x \in A \land x \notin B$

The kids seem kind of into the notation, and the notions. One that caught their attention today was the distinction between a subset, and a proper subset. See how the subset symbol looks a little like “less than or equals”? Like ≤? Well, if we take the line away, then it means a “proper” subset, and it must strictly be smaller than the set, or be missing something that is in the set.

Looking at the examples above, B is a proper subset of A (there’s nothing in B that’s not in A, AND there is something in A that’s not in B), but C is not a proper subset of B. It’s true, C is a subset of B, but there is nothing in B that is not in C, so C is not a PROPER subset of B.

In symbols, B is a proper subset of A, $B \subset A$

C is not a proper subset of B, $C \not\subset B$

At this stage, if students start to “get it” they are prone to argue almost philosophically.

Today was no exception. I sent them to lunch as they debated whether the empty set is a subset of itself AND a proper subset of itself.

This is teaching for fun. Every student is here voluntarily. They have chosen to suffer. We sit and read, and talk and debate, and I jump out of the circle to run to the board. One by one the kids are getting tough ideas or tricky language.

I’m assigning homework, and taking attendance. But the homework is for class discussion. I am not checking it. I am not grading it. I am not giving tests or quizzes. I will grade my students on the quality of their discussion, which has been uniformly high.

Hanging out and talking about math with kids who want to talk about math. This is teaching for fun.

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