Fun set theory for kids

Question

Are there some fun results in set theory to set as landmarks while introducing to kids?

For example, while introducing graph theory to kids, I could explain isomorphism via a pentagon and pentagram, introduce the Eulerian graph via the 7 bridges puzzle, show $V+F-E = 2$ as it is fun itself, and explain the Euler characteristic via drawing a utility graph on a coffee cup. These are funny results, but also linked up as an interesting journey.

Are there similar funny results to show the basics of set theory? The barber paradox could be one, but it's something to explain "why we need proper defined set theory", other than "real set theory stuff after drawing the boundary".

Answer 1

Hilbert's Hotel is a nice thought experiment for explaining results about cardinality of infinite sets and the aleph numbers.

I have also used plastic bags to explain the difference between $\varnothing, \{\varnothing\}, \{\varnothing,\{\varnothing\},\{\{\varnothing\}\}\}$ etc. to kids. Let an empty plastic bag represent the empty set. Then a plastic bag containing only an empty plastic bag is $\{\varnothing\}$, and so on.

Answer 2

I found the formula connecting the union and intersection of two sets useful at school.
$$n(A\cup B) + n(A\cap B) = n(A) + n(B)$$

Say you wish to find how many numbers from 1- 1000 inclusive are multiples of 10 or 25. This may be phrased as which money amounts up to $10 can be made just from dimes or just from quarters.

Let A be the set of amounts that can be made from dimes, there are 1000/10 = 100 of these.
Let B be the set of amounts that can be made from quarters, there are 1000/25 = 40 of these.
The amounts that can be made from both are multiples of 50, so there are 1000/50 = 20 of these.

The answer is then:
$n(A\cup B) + n(A\cap B) = n(A) + n(B)$
$n(A\cup B) + 20 = 100 + 40$
$n(A\cup B) = 120$

Answer 3

Questions about infinity are one way to go. e.g. 'Are there more natural numbers or even natural numbers?'

Intuition says there are more natural numbers ($\mathbb{N}$) than even natural numbers ($2\mathbb{N}$), but division by two gives a bijection from $2\mathbb{N} \to \mathbb{N}$.

It is counterintuitive enough to drive discussion and really puts a focus on definitions. For example, you could ask someone who disagrees to put forth another definition of size and contrast it with cardinality.

Answer 4

A great way to get folks engaged is to bring up a contentious question. Here's one:

What's the 0th power of a whole number? In particular, what's the 0th power of 0?

Like Lorenzo Najt mentioned, you can settle this pretty convincingly if you treat numbers as abstractions of finite sets. I think my write-up at Math.StackExchange is pretty kid-friendly, and could be made even more so.

Answer 5

Is it possible to have a bag of stones, and every time you arrange them on the ground in a spiral and count them, you get 64, but whenever you arrange them to fill a triangular region and count them then you get 63?

You can play the devil's advocate until the child gets their proof in shape. ("No, that's impossible!" "Well, why would it be impossible? It's easy to imagine." "But it has to be the same number!" "What does that mean? Why would it have to be the same?" etc.) When their argument is solid, you capitulate and they enjoy their success.

It helps with thinking about bijections and the meaning of equal-sized sets, and it also helps convey the feeling that proofs are how we can really be sure of things (even if the word "proof" will wait until they are older).

Answer 6

Richard Schwartz's picture book Life on the Infinite Farm illustrates three funny bijections between infinite sets. The first two are the classic bijections from $\{\bullet\} \sqcup \mathbb{N}$ and $10\mathbb{N}$ to $\mathbb{N}$. The third is a mind-bending bijection from the edges of an infinite binary tree $B$ to the edges of $B \sqcup B$.

The pictures, like the bijections, are zany, colorful, and a little disturbing. I really like the narrative's pacing. It sets up three conflicts that leave you wondering, "How's our pal gonna wriggle out of this mess?" Then it lets you stew for a while before the bijection swoops in and saves the day. It reminds me of what Dan Meyer says about aspirin and headaches—though with infinite sets, the aspirin is often a headache of its own.

Answer 7

Depending on what you mean by 'kids':

Maybe you could explain the formulas $|A \dot{\cup} B| = |A| + |B|$ and $|A \times B| = |A||B|$ and $|Maps(B,A)| = |A|^{|B|}$.

This let's you see all sorts of arithmetic identities as identifications between sets of maps, which can be more intuitive than pure arithmetic (or differently intuitive).

For instance $|A|^{|B| + |C|} = |A|^{|B|} |A|^{|C|}$ would come from thinking about how to define functions on a disjoint union, and $(|A|^{|B|})^{|C|} = |A|^{|B||C|}$ from thinking about how $A$ valued functions on a $B \times C$ are the same thing as $Maps(C,A)$ valued functions on $B$ (Is it ever too early to learn about currying?). You can also explain the distributive property this way.

Probably these identities best posed as exercises to think about over a long period of time, but I'm not sure of how to pose them as riddles that don't feel forced...

A related thing to discuss would be $\mathscr{P}(A) = Maps(A, \{0,1\})$. You can use this to explain why $2^n = \sum_{k = 0}^n {n \choose k }$... related to this, another place where elementary set theory shines is in doing combinatorial calculations in a careful and unambiguous way - reasoning on the level of words gets confusing for me, its better to build sets and prove relationships among them.

Answer 8

1. Starting with some mind blowing paradoxes like asking the famous Russel's set paradox- does a set, that consists of all the sets that don't consist of themselves, consists of itself? (If it does - it doesn't, if it doesn't - it does).
2. Establishing the term of bijective functions, and with finite examples explaining the motivation to define two equivalent sets as two that have a bijection between them.
3. A very cool and intuitive definition for an infinite set - a set which is equivalent to a strict subset of itself. It is nice to demonstrate with the Natural numbers and the Even numbers.
4. After the bijection term is well-established, it is time for Cantor's Diagonal Argument, to show that there could not be a bijection from the Natural numbers to the unit segment. I think this might just be the best 'party trick' to use when you want someone to get enthusiastic about set theory, though I find it necessary to talk a bit about bijections first.