How to Solve a Math Problem - At A Glance

There are three steps to solving a math problem.

1. Figure out what the problem is asking.
  
2. Solve the problem.
  
3. Check the answer.

Sample Problem

The Cantor set, denoted c, is a particular subset of the interval [0,1].

Our favorite way to describe c is as the limit of a particular sequence of sets C₀, C₁, C₂, C₃, ....

Let C₀ be the unit interval [0,1]:

To get C₁, remove the open interval  from the middle of C₀:

The set C₁ consists of two line segments, each of length .

To get C₂, remove the open interval that forms the middle third of each of those line segments:

The set C₂ consists of four line segments, each of length .

We keep going this way for all eternity, and define the Cantor set by

This means c consists of the points that are in all the sets C_n.

Let the set C be the complement of the Cantor set - that is, all points in [0,1] that aren't in c.

Find the length of C. What does this tell us about the length of c?

Answer.

1. Figure out what the problem is asking.

There are a lot of words here, but what's really going on?

We start with the interval [0,1]:

We're going to take out a bunch of stuff and throw everything we take out into the set C.

Every time we remove the middle of a line segment, the portion we remove goes into C:

The question is asking us to find the length of C. In other words, it's asking for the sum of the lengths of all the line segments we remove along the way.

2. Solve the problem.

We need to add up the lengths of all the line segments we remove. On the first step, we remove one line segment of length .

On the second step we remove two line segments, each of length . The sum of their lengths is .

On the third step we remove four line segments, each of length . The sum of their lengths is .

At this point we can see the pattern:

The total length of the line segments we remove from step 1 to step n is

In the limit, the total length of all the line segments we remove is

This is an infinite geometric series with  and , so its sum is

The total length of all the line segments we remove is 1. This means the length of C is 1.

Since C is everything that's not in c, this suggests that the length of c is 0.

3. Check the answer.

We don't have a great way to check the answer to this one, but we can at least think about whether it makes sense.

We found that the length of c is 0. This means c can't contain any line segments at all.

Well, c certainly doesn't contain any line segments longer than  because after step 1 the longest line segments remaining have length . c also doesn't contain any line segments longer than  because after step 2 the longest remaining line segments have length

.

Continuing like this we realize c can't contain any line segments longer than , for any value of n.

This means that c can't contain line segments of any length greater than 0. Consider any non-zero length L. There's a value of the form  that's smaller than L, which means L is too big for c to contain any line segments of that length.

Since no line segments of any non-zero length exist in c, it makes sense to say that "the length of c is 0."

Although the length of the Cantor set c is 0, it does contain points. The Cantor set contains, at the very least, the points 0, 1, , and , so we can see that it's not empty. In fact, it turns out that the Cantor set contains uncountably many points.

The Cantor set has length 0, but uncountably many points. Does your brain hurt yet?