Even More on the Cantor Set


Recall that we have set up a correspondence between the points of and the collection of all sequences of L's and R's.

But what's so special about L's and R's? Not a thing. If we can agree to use sequences of 0's and 1's instead, then it's nearly immediate that is a BIG set! Why? Well, a sequence of 0's and 1's

011010001110010...

looks just like a binary decimal

0.011010001110010... (base 2).

Now each point in [0,1] has a binary decimal expansion and, conversely, each binary decimal represents some point in [0,1]. Thus, we have a correspondence between the points in the Cantor set and the points in the interval [0,1]. Amazing!!

In truth, there are some details to check here. For example, a given element of [0,1] may have more than one binary decimal expansion; indeed, 1/2 can be written as

1/2 = 0.1 (base 2) = 0.01111111... (base 2).

(How would you check this?)

The key observation here is that we can specify a mapping, or correspondence, from onto [0,1]. Since is a subset of [0,1], this means that [0,1] and must be the same "size" (or have the same cardinality) -- but let's save the actual details for another day.

Is this cool or what? A set so small that we couldn't possibly draw it on paper (or, in this case, on a computer screen) is still, somehow, as big as an entire interval.

Here's that "big versus small" deal again: The Cantor set is apparently "big", since it has the same cardinality as the interval [0,1] (an uncountable set). If you stick around to read yet more on the Cantor set, you would see that the Cantor set is pretty darn small, too. Of course, if you're tiring of magic and wonder, you could always wimp out.


Neal Carothers - carother@bgnet.bgsu.edu