Sets as Boxes

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Sets as Boxes is a simple visualization tool
for students of any age who are first encountering naive set theory.

Having trouble with the difference between the empty set {} and the set {{}}? Can't seem to figure out how many elements {{{}},{100}} has? This program may help clarify things for you.

Think of a set as a cardboard box. It may have nothing in it, or it may have some things in it, just like cardboard boxes do in everyday life. In fact, it might even have other cardboard boxes in it. As you might expect, Sets as Boxes uses the empty box to represent the empty set. The empty box is shown at the top of this page. Here are some other examples.

A box within a box,
the set {{}}.

A box containing two balls,
sometimes called "atoms" in set theory because they're not sets.
A set containing two numbers.

Sets as Boxes provides the following features for visualizing simple sets like these, and slightly more complex ones.

  • Keyboard input of standard mathematical notation using { } , and digits 0-9.
  • Pre-packaged common example sets (the first few natural numbers, ordinals, and some example Kuratowski pairs).
  • Validation of your input and explanation of any typographical errors you made.
  • 3D visualization of your input as boxes containing other boxes and balls and/or numbers.
  • Clicking a subexpression of the expression you entered causes Sets as Boxes to raise up that box to distinguish it. See screenshots below for an example.

Example screenshots appear below. They were captured on a Macintosh, but Sets as Boxes is distributed for Windows XP, Mac OS X, and as source code for compilation on other platforms.

Some of the examples in the popup menu available. If you're interested in any of the example sets given, this will save you some typing.
A view of the full window of Sets as Boxes. The user has clicked on the set {2} and highlighted it, and thus the box containing 2 has been raised up above the other boxes in the 3D view.
Author: Nathan Carter