Some summary background on cardinality and countably infinite sets


We decided in class that two sets have the "same size" if there is a one-to-one correspondence between them:
For example, the following shows a one-to-one correspondence between a set of barnyard animals and the set {1,2,3,4,5,6}:

          
The important things in making a one-to-one correspondence is that every member of one set has one (and no more than one) partner in the other set.
For example, the next diagram is not a one-to-one correspondence; even though such a correspondence exists, this isn't one of them:

         
This fails to be a one-to-one correspondence for two reasons:  "1" has no partner at all, and "2" has more than one partner.

Of course, there are simpler ways to make a valid correspondence than our first example:

         

As we will see, with finite sets, making a correspondence is always easy.

So we make a definition: Two sets have the same cardinality if there is a one-to-one correspondence between them.  In the case of finite sets, we can think of cardinality as being the number of members a set has.

How does this idea of same cardinality play out with infinite sets?  First, some further definitions:

We say a set A is finite if there is a natural (or "counting") number n so that A can be put in one-to-one correspondence with the set {1,2,3,...,n}.

Our example above shows that our set of barnyard animals is finite, since it can be put in one-to-one correspondence with {1,2,3,4,5,6} - so the "natural number n" in our definition above would be 6 in this case.
Likewise, the set {Ford, Smullin, Eaton, Waller, Collins} is finite because it can be put in one-to-one correspondence with {1,2,3,4,5} - so the "n" in this case is 5.

We say that a (nonempty) set A is infinite if it is not finite.  (the "nonempty" is to avoid a technicality).
So our prototypical infinite set is the set of all natural or counting numbers, {1,2,3,...}, which we denote by the symbol [bold N].

It is easy to see that the negative integers has the same cardinality as [bold N]:


(note this is "only part" of the real diagram, since the "real thing" contains infinitely many members on each side, and infinitely many connecting lines; the diagram really tells us how to make the correspondence: every counting number is paired with its opposite.)
You should take a moment and persuade yourself that this diagram tells us a correspondence, that is, that every member of either set has exactly one partner in the other set - no member is left alone, no member has two partners.

A more troubling example:


This shows that {-3,-2,-1,0,1,2,3,...} has the same cardinality as {1,2,3,...}. 
    First, one should check that this really is a one-to-one correspondence: does every member on the left have a partner on the right?  Yes, each number on the left is partnered with the number 4 less on the right, so 7(on the left) is partnered with 3(on the right), 4 is partnered with 0, and so on.  What will 15(on the right) have for a partner?  19(on the left).  Nobody is left lonely, and nobody gets two partners.
    Next, let's explore why this is troubling: the right hand set, {-3,-2,-1,0,1,2,3,...}, has all of the members of {1,2,3,...} plus four more (namely 0, -1, -2, and -3), yet somehow both sets have the same cardinality or "size."  We can dismiss this problem by saying "both are infinite, hence both the same size," but as we saw in class, this isn't necessarily true.

Some other terminology:  A set A is countably infinite if it can be put in one-to-one correspondence with [bold N] (remember that's {1,2,3,...} ).
So the examples above show that the negative integers and the set {-3,-2,-1,0,1,2,3,...} are both countably infinite.

Even more surprising (perhaps): The set of all integers (both positive and negative and zero too) is countably infinite, i.e. can be put in one-to-one correspondence with [bold N].  Now it was easy to "count" an infinite set which has a "first" member and then a "next" member, and so on, such as {-3,-2,-1,0,1,2,3,...}, but the set of all integers doesn't have a "starting point" per se.  Yet we can still make the one-to-one correspondence with [bold N]:

   
Again, this diagram really only suggests the real correspondence, which has infinitely many "lines", but the above is already messy enough to make it hard to read.  The idea is we start corresponding 1(on the left) to 0(on the right), then alternate corresponding numbers on the left to positives and negatives on the right.  To make sure we can continue this correspondence, we should be able to say how it works in general:  Every odd number n(on the left) corresponds to (n-1)/2 on the right, so for example 5 corresponds to (5-1)/2 = 2.  Every even number n(on the left) corresponds to -n/2 on the right, so for example 8 corresponds to -8/2 = -4 on the right.  One can check that every member on either side has one and only one partner; for example 51 on the left corresponds to 25 on the right; -14 on the right corresponds to 28 on the left, 8 on the right corresponds to 17 on the left.

Beware the temptation to dismiss this surprising result (surprising because somehow it seems that "the integers have at least twice as many members as [bold N]") by saying "well, both sets are infinite after all, so they are the same.  Again, as we saw in class, not all infinite sets have the same cardinality.

Homework:
  1. Let E be the set of positive even integers, so that E={2,4,6,8,10,...}.  Show that E is countably infinite, i.e. E can be put in one-to-one correspondence with the natural numbers, i.e. {1,2,3,...}, by drawing a diagram and giving a few sentences describing the correspondence in general.
  2. Let F2 be the set of all positive fractions whose denominators are 1's or 2's, so F2 contains all of 1/1, 2/1, 3/1, 4/1, ... and all of 1/2, 2/2, 3/2, 4/2, 5/2, ... .  Show that F2 is countably infinite.  Beware:  remember that 4/2 and 2/1 are the same; say a few words about how you avoid "double counting," i.e. giving some of the counting numbers in [bold N] more than one partner in F2.
  3. Review (or think about) how you would put the set F of all positive fractions in one-to-one correspondence with [bold N].  It will be helpful to write F in an array:
                          1/1   2/1   3/1   4/1   5/1   6/1   ...
                          1/2   2/2   3/2   4/2   5/2   6/2   ...
                          1/3   2/3   3/3   4/3   5/3   ...
                          1/4   2/4   3/4   4/4   5/4   ...
                          1/5   2/5   3/5   4/5    ...
                          1/3   2/3   3/3   4/3    ...
                           .     .     .     .     ...
                           .     .     .     .     ...
                           .     .     .     .
    Beware: This array repeats a great many of the fractions, for example 1/2 appears as 1/2 and as 2/4 and as 3/6, etc. - describe how you deal with them.