What is an example of something you might see outside of math class and how would you model that thing as a set?

Question

In mathematics, we have sets, such as $\begin{Bmatrix}1, 2, 3 \end{Bmatrix}$ or the real-numbers, usually denoted as $\mathbb{R}$.

When teaching students about sets for the first time, it can sometimes (not always, but sometimes) be nice to make an semi-intuitive analogy to real-life.

What collection of 3 to 10 words in the English language can we use to represent a mathematical set?

Ideally...

• There exists two sets $A$ and $B$ in our example such that $A$ and $B$ overlap ($A \cap B \neq \emptyset$)

• There exists a universal set labeled $(\alpha\Omega)$ in our example such that $A \subseteq (\alpha\Omega)$ and $B \subseteq (\alpha\Omega)$ and $A \cup B \neq (\alpha\Omega)$

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We could ask students for such an example, except that we are teaching student who do not already know what a set is.

These students have never heard the words "union" or "intersection" used in math-class before.

Our goal is to choose a handful of words from every day English in order to provide an analogy for what a set is.

Someday, I would like to present students with many examples of this, but I am having difficulty thinking of more than three examples.

Answer 1

Examples of Sets You Might see Outside of Math Class

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Example Using Coordinates of Wolves in a Video-game

A person could could imagine having a RPG (real-time role-playing) videogame with $\mathcal{wolves}$

The geo-special coordinates for every animal in a videogame might be stored in a set named $\mathcal{animals}$.

There are more than just $\mathcal{wolves}$ in this videogame. Students can be asked to suggest what other animals they might want in the videogame. It is important to me that students participate actively.

Not every animal in the videogame is a $\mathcal{wolf}$ Thus, the set of coordinates for $\mathcal{wolves}$ is a subset of the set of GPS coordinates for all $\mathcal{animals}$.

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Example Two of a Set Outside of Math Class

In our second example, the set of all geo-spatial coordinates of cups, plates, forks and other dishes in someone's kitchen might work. The set of all water glasses is a subset of the set of all drinking vessels. You could also have asian tea bowls, clay coffee mugs, and other subsets of the set of drinking vessels.

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Example Three of a Set Outside of Math Class

We could take $\mathcal{electronic\_devices}$ to be the universal set in this example. We let $\mathcal{disp\_devs}$ be the set of all devices which have a display which shines out light for your eyes to see. The set of all $\mathcal{newerphones}$ would be a subset of the set of all $\mathcal{disp\_devs}$. We could show students pictures of old fashion telephones from museums.

Answer 2

The best colloquial word synonym for set, I've heard is "collection". Not in the sense of a stamp collection, but just in the sense of collecting anything. And it's just for introducing the idea, not for rigor (not saying every set is a physical collection of objects). I think Tenenbaum and Pollard use that idea to remind kids of what sets are, in their ODE primer...and it works nicely.

I would try to keep it very simple at first. For example for the set of all chairs in your house, the kitchen chairs would be a subset. Or the intersection of all chairs in the house with all furniture in the kitchen is...the kitchen chairs.

All that said, I think the idea of union and intersection is pretty easy to get and I was fine with collections of letters or numbers to start with the idea. I think this is an easier access than entire number systems like integers or reals or whatever, at least at first. I find the whole R or C (special fancy font) slightly offputting.

Answer 3

The example which is frequently used in textbooks is the set of students in a school who may be taking a Foreign Language class or taking a Social Studies class. You could also talk about the students who are taking either type of class, and have them talk about the fact that some people might be taking both. Talking about students taking Spanish class vs. any language would naturally lead to the discussion of subsets.

The benefit of this example is that it is something the students have intimate familiarity with.

Answer 4

I would just use English pronouns (classical ones, though, if you are in the mood for it, you can add some Canadian neologisms too, provided that you can figure out how to pronounce them and remember which one is which). It is a simple not too large universe with several clearly defined subsets:

singular/plural

masculine/feminine/neuter

first/second/third person

subject/object/possessive

Starting with "i","y","t",etc.

1 letter long/2 letters long/3 letters long, etc.

You can make a full list of pronouns you want to use and distribute it to the students so that you all agree on what the underlying universe is, after which everything becomes well-defined and you can discuss all the set-theoretic concepts on simple examples.

Answer 5

Couldn't leave a comment due to rep

I just wanted to elaborate on:

"These students have never heard the words "union" or "intersection" used in math-class before."

There is a strong relationship between logic and set theory. You can make use of that. The key assumption is that they have studied logic before.