21

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 ...


11

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 ...


9

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 ...


5

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, ...


4

The US has different courses at different schools. Sometimes with same name but differences in content or prereqs. It would be more meaningful to sketch this tree for a given school. Or do a few schools. That should give you some feel for the general lay of the land. And I suggest to sketch it yourself, using a course catalog. You'll learn more doing a ...


4

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, ...


4

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'...


2

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|^{|...


2

This seems like a quite peculiar assortment of topics, and I do not imagine that any existent text addresses all of these. I think you will need to either write your own materials (if you want a coherent treatment), or just cobble together textbook treatments from a diversity of sources.


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). Establishing the term of bijective functions, and with finite examples explaining the motivation to define two equivalent ...


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