22 votes
Accepted

Fun set theory for kids

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 $\...
A. Goodier's user avatar
  • 1,715
21 votes
Accepted

Can this be a better way of defining subsets?

In both formal and informal treatments of set theory, we need to specify which operations and relations are allowable and build from there. Usually we take sets and set membership as primitives. We ...
Steven Gubkin's user avatar
14 votes

How can I convince my brightest student of Cantor's theory?

Not every property is preserved by limits. Here is a more basic situation in which the same reasoning is used: For each natural number $n$, there are only finitely many natural numbers in the ...
Steven Gubkin's user avatar
13 votes

Why is set theory not taught at the outset of math education?

Not a complete answer (could there be one?), but too long for a comment. why is it [Set Theory] not being taught at the very outset of math education? It has been tried, most likely still is in ...
dxiv's user avatar
  • 271
13 votes

Can this be a better way of defining subsets?

In addition to the very good answers that you've already received, it's probably worthwhile to also mention the following point: The alternative that you suggest might lead to a similar type of ...
Jochen Glueck's user avatar
11 votes

Fun set theory for kids

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 ...
Tom's user avatar
  • 333
11 votes
Accepted

Is it considered a mistake to use different correct notation for writing intervals?

The goal of notation is to clearly and precisely communicate an idea. The notation $$ (a,b) = \{ x : b < x < a \} $$ fails to clearly communicate an idea, as this notation is unusual, and ...
Xander Henderson's user avatar
  • 7,698
9 votes

How to motivate equivalence classes

As a simple illustration of the most basic case, $\mathbb{Z}/2\mathbb{Z}$, I sometimes push the light switch of the classroom about a dozen times in a row, too fast for them to count. Then I ask: &...
Benoît Kloeckner's user avatar
9 votes

How can I convince my brightest student of Cantor's theory?

The problem is not one of reconciling analysis and set theory at all; it's one of understanding analysis properly (and how things go sideways when dealing with the infinite). One of the first and ...
rnrstopstraffic's user avatar
9 votes

Fun set theory for kids

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\...
Adam's user avatar
  • 5,382
8 votes

Cognitive traps in very early set theory

I think your specific situation with the exam question is similar to a pre-Calc class that spends a good amount of time on solving trig equations, but then asks on the exam for the student to write ...
Aeryk's user avatar
  • 8,025
8 votes

How can I convince my brightest student of Cantor's theory?

Here is another attempt. Consider the function $f:\mathbb{N} \to \mathbb{R}$ defined by $f(n) = \textrm{ The number of elements in the set $\{n,n+1,n+2,...2n\}$}$. In other words $f(n) = n+1$. ...
Steven Gubkin's user avatar
8 votes

Can this be a better way of defining subsets?

This answer is going to have a lot of overlap with Steve Gubkin's, but maybe a bit of a different focus. I agree with him that (1) The key issue here is understanding how vacuous cases work. (2) ...
David E Speyer's user avatar
7 votes

Can this be a better way of defining subsets?

A potential objection: what does it mean to be able to make a set by removing elements from another set? Some students will think of removing the elements one at a time. Then there'll be ...
kaya3's user avatar
  • 530
6 votes

Determining sets to show sufficiency of a condition?

I would not recommend to teach this method since there are some downsides. Take $A(x) \iff x \text{ is divisible by } 2$ $B(x) \iff x \text{ is divisible by } 42$ Is $A(x) \implies B(x)$ or $B(x) \...
Stephan Kulla's user avatar
6 votes

Determining sets to show sufficiency of a condition?

First, although you talk a bunch about cardinality, I don't see how that makes sense, so I'm going to assume you mean that you have them determine if the set corresponding to p is a subset of the set ...
Henry Towsner's user avatar
6 votes

Cognitive traps in very early set theory

I think the issue here is really more one of problem solving skill or abstraction ability. You could see the same thing in various problems in computer science or in college algebra where some ...
guest's user avatar
  • 81
6 votes
Accepted

Category mistakes regarding symbols and their impact on math (mis) understanding. ( Object symbol/ sentence symbol confusion)

I like the idea here, but I agree that it misleads students, and might have the opposite of intended effect. Why not hand out a paragraph to the students, and ask them to critique it. Say that the ...
Steven Gubkin's user avatar
6 votes

Mnemonics to correlate the definition of "asymmetric relation" and "antisymmetric relation" with the terms

First, let's note that the terms as used by Rosen are standard definitions, as we can see on Wikipedia (here and here), as well as other resource sites. There was some question about this in the ...
Daniel R. Collins's user avatar
6 votes

Is it considered a mistake to use different correct notation for writing intervals?

In theory you could choose to define a notation such as $$(a, b) = \{x ; b < x < a\},$$ but why would you do it? It has exactly the same power as the usual notation, and now the reader either ...
Tommi's user avatar
  • 7,009
6 votes

Can this be a better way of defining subsets?

The only flaw is that it doesn't really fix the perceived "problem" of the original definition. What if $A$ and $B$ are both empty? What elements of $B$ are you going to remove to get $A$? I ...
MPW's user avatar
  • 161
5 votes
Accepted

When self teaching, should I learn set theory before continuing ap calculus?

Short answer: No. If you really want to know a little bit about sets, the first section of Book of Proof or Applied Discrete Structures would more than suffice for the purposes of most collegiate ...
kcrisman's user avatar
  • 5,980
5 votes

Why is set theory not taught at the outset of math education?

Because many worthwhile topics in math don't require set theory to learn them and apply them. You can get a lot of good stuff done with arithmetic, algebra, trig, calc, etc. that is uninformed by set ...
guest's user avatar
  • 277
5 votes

Fun set theory for kids

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, ...
Vectornaut's user avatar
4 votes

Where can I find the partial order relation of prerequisites of undergraduate courses in the United States?

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 ...
guest's user avatar
  • 41
4 votes

Cognitive traps in very early set theory

I may be wrong, but from personal experience I think the other answers are missing a crucial point, what it is like to be taking the test itself, and how people attempt to learn material. If the ...
Krupip's user avatar
  • 291
4 votes

How to motivate equivalence classes

To be honest, I wasn't sure to write this answer or not, since it was hard to choose what to write about! Believe or not, my whole PhD thesis entitled "Equivalence" was to provide a ground for ...
Amir Asghari's user avatar
  • 4,428
4 votes

Fun set theory for kids

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 ...
Matt's user avatar
  • 141
4 votes

Whole numbers as sets vs abstracted properties of sets

Disregarding research on the foundations of mathematics, what we really want to have are the usual numbers with the usual properties. The usual properties are really nice, because they correspond to ...
Tommi's user avatar
  • 7,009
3 votes

Cognitive traps in very early set theory

I don't think I can address cognitive traps, but perhaps this tool could be used for student practice. There is a "Wolfram Alpha Widget" for exploring set Venn diagrams. The widget code is available ...
Joseph O'Rourke's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible