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".

  • $\begingroup$ There are lots of amusing Venn diagrams out there. Maybe they could make their own Venn diagram jokes. $\endgroup$ – Sue VanHattum May 4 '20 at 20:00
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    $\begingroup$ Do you mean funny as in in amusing or fun as in enjoyable, or something else? Funny can also mean strange. $\endgroup$ – J W May 4 '20 at 20:33
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    $\begingroup$ @JW sorry for my poor English, by funny i mean the kids would enjoy, find it interesting, puzzled at first then have an "aha" moment... $\endgroup$ – athos May 5 '20 at 8:51
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    $\begingroup$ What age are the kids, though? $\endgroup$ – Robbie Goodwin May 5 '20 at 22:12
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    $\begingroup$ What target age are we talking about here? "kids" and "set theory" call two very different age groups to mind for me, but maybe not for you. $\endgroup$ – Flater May 6 '20 at 11:07

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.

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    $\begingroup$ Empty plastic bag for $\varnothing$ is a great idea, intuitive! $\endgroup$ – athos May 4 '20 at 16:28
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    $\begingroup$ Nice, although it does break down on {∅,∅}, suddenly your kid is going to class, telling everyone that set theory proves that a bag with two bags is the same as a bag with a single bag :-p $\endgroup$ – epa095 May 5 '20 at 10:46
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    $\begingroup$ @epa095, it is the same unless you switch to using multisets, but I see the problem. $\endgroup$ – J W May 5 '20 at 11:05
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    $\begingroup$ @epa095 I still think the bag/box analogy works well for the empty set. {∅,∅} is no problematic than {1,1} under that analogy. $\endgroup$ – Nico Burns May 5 '20 at 23:03
  • $\begingroup$ Maybe let kids play as "collectors of unique/original items"? You keep collections in bags, and more than 1 of the same item is considered worthless in your club :-). When your club started getting bored, someone proposed collections themselves should be unique; then someone invented "collections of collections"! Give each kid 3 tokens and 3 bags, and let them discover original ideas... Ideally, let them discover the empty set by themselves! $\endgroup$ – Beni Cherniavsky-Paskin May 7 '20 at 11:07

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

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    $\begingroup$ yes quite practical, thx! $\endgroup$ – athos May 4 '20 at 16:28
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    $\begingroup$ I like going the practical route as well. I majored in math, taught high school math, and still learn math for fun, and I've always thought the appeal of things like cardinality and the Banach-Tarski paradox are WAY overrated in general, and particularly on this site. If as a young student I was told that a fully booked hotel can fit another guest (it can't by definition of fully booked) or that a ball can be split into any number of identical balls (it can't) I would not be able to take the speaker seriously. $\endgroup$ – Thierry May 4 '20 at 16:44
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    $\begingroup$ But don't get me wrong: I still think Hilbert's Hotel is a cool way to visualize some results, and it's cool that a mathematically idealized ball behaves weirdly, but as an introduction for kids I don't see the appeal. Basic results that can be visualized with Venn diagrams seem like the way to go. $\endgroup$ – Thierry May 4 '20 at 16:51

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.


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.

  • $\begingroup$ The operations "raise X to a whole number" and "raise X to a member of the same field" are semantically different operators. They may behave identically when X is a member of the algebraic ring of whole numbers, but the behavior with real numbers is different if X is zero. $\endgroup$ – supercat May 7 '20 at 21:27

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).

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    $\begingroup$ That's a nice one. Maybe something similar can be done with a big bag of water bottles -- "without counting, are there at least as many caps as water bottles?" "yes, because nothing is spilling." ... maybe there's a related example where similar thinking can guarantee a bijection? $\endgroup$ – Lorenzo Najt May 5 '20 at 3:07

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.

  • $\begingroup$ thx the recommendation. i love such popular science books, professionals showing not only the knowledge but their insight, not to mention the eye catching comic! $\endgroup$ – athos May 5 '20 at 13:33
  • $\begingroup$ This book is awful, I would scare of math if I saw this pictures at age bellow 10. $\endgroup$ – Aqua Aug 7 '20 at 9:16

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.

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    $\begingroup$ Currying is a great idea, from number to variable, to function, to functor.. thx! $\endgroup$ – athos May 5 '20 at 9:25
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    $\begingroup$ @athos Maybe Linderholm's "Mathematics made difficult" could be a source of inspiration as well. (This is book for mathematicians, not kids, but you might be able to extract some fun games or examples from it.) $\endgroup$ – Lorenzo Najt May 5 '20 at 22:44
  • $\begingroup$ @athos The write up linked by vectornauts answer contains some kid friendly ways to motivate and define disjoint union, cartesian products, and sets of maps. $\endgroup$ – Lorenzo Najt May 6 '20 at 15:10
  • $\begingroup$ @athos I was mainly referring to describing Hom(X,Y) as Y-colorings of X. $\endgroup$ – Lorenzo Najt May 6 '20 at 21:25
  • $\begingroup$ ah now I see, yeah it's quite nice to have a tangible explanation on an abstract idea. $\endgroup$ – athos May 6 '20 at 22:08
  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.
  • $\begingroup$ I believe most of it would work with 7th graders and beyond $\endgroup$ – Eliyahu Abadi May 7 '20 at 12:14

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