Stat 61
10/7/2007
**A Small Essay on Countable Sets**
**What do “countable” and “uncountable” mean?**
Infinite sets come in two varieties: countable (or “countably infinite”) and uncountable. An infinite set is __countable__ if its members can be listed in a sequence (or “list”) indexed by the positive integers 1, 2, 3, …, like this:
A = { a_{1}, a_{2}, a_{3}, … }
The point is that positive-integer subscripts can be assigned to all of the members of A. Any way of assigning subscripts suffices, as long as every member of the set eventually gets a subscript.
**What are some examples?**
Obviously, the positive integers themselves, called Z^{+} or N, are a countably infinite set. Just let each integer be its own subscript. Or, you can mix them up in any way you like. (For any countably infinite set: Once you find one way to assign subscripts, you’ll find that there are many other ways.)
The whole set of integers, Z, is also countable:
Z = { 0, 1, -1, 2, -2, 3, -3, … }
That is, Z = { a_{1}, a_{2}, a_{3}, … } where a_{n} = n/2 if n is even, and -(n-1)/2 if n is odd. It is obvious that every integer, whether positive or negative, eventually appears in this list.
What about the set of integer lattice points? Write:
G = { (x, y) | both x and y are integers }.
Then G is countable. A picture tells the story. We can write:
G = { (0,0), (1,0), (1,1), (0,1), (-1,1), (-1,0), (-1,-1),
(0,-1), (1,-1), (2,0), (2,1), (2,2), (1,2), … }
where the pattern is given by the picture. It’s a lot of trouble to give a formula for the subscripts — way more trouble than it’s worth — but it’s obvious that every lattice point will eventually appear on the list.
What about the set of rational numbers? Every rational number can be written in the form p/q where p and q are integers. So, we can associate it with the integer lattice point (p, q) in the previous example. When we construct the list, we have to skip some of the pairs… like (1, 0), of course, since 1/0 isn’t a rational number, and also pairs like (3, 6), since 3/6 is the same rational number as 1/2 and has already been listed. But we can still construct a list by this method, so the rational numbers are a countable set.
**What are some examples of sets that ****aren’t**** countable?**
The set of real numbers is uncountable. So is any interval of real numbers (except singletons, of course, and the empty interval).
The proof isn’t obvious, but it’s very beautiful. [Skip it if you like. But come back sometime.]
Consider the interval half-open interval [0,1). We have chosen this interval for convenience, because every number in the interval has a decimal representation that starts with “0. …” and any decimal representation that starts with “0. …” represents a number in the interval.
We’ll prove that the numbers in this interval cannot be written as a sequence, indexed by the positive integers.
We’ll prove this by contradiction. Suppose I claim to have formed a list (=sequence) of numbers in the interval, with subscripts from Z^{+}:
r_{1}, r_{2}, r_{3}, r_{4}, …
Then you can find a number x which is not on my list, by specifying all its digits, as follows.
(a) If the first digit (to the right of the decimal point) of r_{1} is “2”, then the first digit of x is “3”. Otherwise, the first digit of x is “2”.
(b) In general, if the n-th digit if r_{n} is “2”, then the n-th digit of x is “3”, and otherwise the n-th digit of x is “2”.
For example, if my sequence is
e-2, pi/6, sqrt of (1/2), 1/19, …
(or, roughly: 0.__7__1828…, 0.5__2__360…, 0.70__7__11…, 0.052__6__3…, … )
then your number is
x = 0.2322…
To find more digits of x, you would only have to see more terms of my sequence.
Now, behold: Your number x is not on my list. That’s because
(a) x is different from r_{1} because their first digits are different.
(b) x is different from r_{2} because their second digits are different.
(c) In general, x is different from r_{n} because their n-th digits are different.
So, x is different from every number on my list. My list didn’t include all the numbers in [0,1) after all. Since you can do this with any list, we conclude that it is impossible to arrange the numbers in [0,1) in a sequence indexed by the positive integers. So, [0,1) is uncountable. (End of proof.)
What’s going on is that the real numbers are actually a “bigger” set — a higher order of infinity — than the integers. You can count the integers, but you can’t count the real numbers.
**Are there other examples of uncountable sets?**
Yes, there are uncountably many examples. Since we care about sets of sets (or, to be less confusing, collections of sets), let’s describe some uncountable collections of sets:
The collection of all intervals in R
The collection of all singletons (one-point subsets) in the
interval S = [0, 60)
The collection of all intervals of the form (-x, x) where x is positive
The Cantor set
The collection of all subsets of R^{3}.
**Why does this matter in a probability class?**
The additivity axiom applies only to countable (or finite) collections of sets.
That second example above — the collections of singletons in S = [0,60) — is important to us. In class we constructed a “uniform probability measure” on S, with the probability measure defined by
P ( [a, b] ) = (b – a)/60
for every interval with b ≥ a. Singleton sets are intervals of the form [a, a], so they all have measure (a-a)/60 = 0.
Those singleton sets are pairwise disjoint, and their union is all of S. So, if the collection of singleton sets were finite or countable, then the additivity axioms would apply and we would conclude that
P ( S ) = P ( union of all the singletons )
= Sum of [ P(s) ], for all the singletons s
= Sum of [ lots of zeros ]
= 0.0.
This would be a contradiction, because another axiom forces P(S) = 1.
Fortunately, since the collection of singletons isn’t countable, we don’t have this problem. Continuous uniform distributions are very important in probability. Without the distinction between countable and uncountable sets, we couldn’t have uniform distributions.
(Actually, there’s a worse problem with trying to apply the additivity axiom to uncountable unions. Look at the sum in the “equation” above — it is a sum of uncountably many numbers. We don’t have any theory about how to add uncountably many numbers. Much of calculus was about adding sequences of numbers. But sequences can have only countably many terms, so our theory of addition only allows us to add countable sets of numbers. The sum in the equation above involves an uncountable set of numbers. It looks easy because all the terms are zero, but in a more general example we would not even know what this sum means.)
**Does “countable” mean the same as “countably infinite”?**
All sets are of three kinds: finite; countably infinite; uncountable.
Some mathematicians use “countable” to mean “countably infinite,” and some mathematicians use “countable” to mean “finite or countably infinite.” So, if you want to be absolutely clear, you should only say what you mean: “countably infinite” or “finite or countably infinite.” (Redundancy is not as bad as being misunderstood.)
**What does “discrete” mean?**
In a probability class it means “finite or countably infinite.” But beware: the word “discrete” means other things in other areas of math.
(end)
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