Lab ideas for set theory

Question

The title of the question actually says almost everything there is to say. I cannot think of any programming labs whatsoever that would naturally flow from set theory.

My students will be coding in Scheme (Racket), and will be pretty new at it when we approach this topic. Are there any ways to translate the concepts of set theory into a reasonable lab?

We are covering:

• Basic definitions
• Venn Diagrams
• Union
• Cardinality
• Cartesian Products
• Power Sets
• Bijection (as proof of cardinality)
• Infinite sets $\aleph_0$ and $\aleph_1$
• Drawing a diagonal line to prove $|\mathbb{R}| \neq |\mathbb{N}|$
• Russell's Paradox

Answer 1

This answer will be general, rather than specific to your proposed curriculum. Since you note in a comment that it is set theory that you want to stress over Racket programming, I'll suggest the following.

First, it is difficult to advance students on two orthogonal axes simultaneously without confusing at least some of them. Therefore you want to put the stress on set theory and its beauty and limitations over the programming aspect. If you were using a language that they already knew well, I'd make a slightly different suggestion.

But first, give them a good, traditional, view of set theory and how it relates in general to modes of thought. Any good book, even if it doesn't mention programming at all, could be used, even Naive Set Theory by Halmos. Use programming primarily to illustrate things in a simple way so that the functional programming doesn't take up too much of their thought space - just enough to get the flavor of, for example deep recursive programming.

The programming exercises can be quite simple, but I would choose them in such a way that they also serve as an introduction to complexity theory and the problem of growth. Something simple with Cartesian product and many dimensions can illustrate the growth. It is also possible to write functional programs for things that seem simple but have terrible run-time. For example, a naive version of reversing a list can easily be quadratic if care isn't taken.

I don't know what libraries Racket offers for support of sets, but it might be useful to talk about the development of such a library - building a tool box of simple set operations.

But, in the main, use the programming as a way to reinforce the theory, not so much as an end in itself. Set theory is rich enough and with enough applications to math and CS that you don't really need to do much more.

If you really want to teach FP in Racket, then a different set of exercises that help the functional thought process develop would probably be better than trying to force it into one realm.

Answer 2

I propose for instance for

1. basic definitions
2. venn diagrams
3. Union
4. Cardinality
5. Cartesian products

Using SQL language, why?

When you create a schema database you need to think in entities, relations, boolean algebra.

I mean you can study this concepts oriented to programming field, making schemas and identifying relationships between tables.

For example when you use JOINS operators and UNION and UNION ALL concepts you need to abstract the concepts about diagram venn(when you use left join, right join, etc)

So if your students and you need make a knowledge route about these concepts; SQL is a good option for it

I hope to be useful.....