Set of Odds or Evens Bigger?

Date: 06/14/2001 at 11:04:56
From: Danielle
Subject: Numbers

Which is a bigger set of numbers, odds or evens?

I tried solving this problem counting 0 as an even, but 1 is an odd 
and greater than 0, and 2 is an even that is greater than 1, and 3 is 
an odd that is greater than 2. Do you get the picture?

Please contact me as soon as possible.
Thanks, 
Danielle

Date: 06/14/2001 at 12:51:44
From: Doctor Rob
Subject: Re: Numbers

Thanks for writing to Ask Dr. Math, Danielle.

By "bigger set," the question means the set having the more elements,
not that the elements of one are greater than those of the other.
For example, the set {9} is smaller than the set {2,3,4}, because it
has fewer elements.

The answer is that the two sets are exactly the same size. You can see 
this by putting the odd numbers into one-to-one correspondence with 
the even numbers using the correspondence,

   for every integer n, 2*n + 1 <--> 2*n,

The left side covers all the odd numbers, and the right side covers
all the even numbers.  Every odd number has a unique corresponding
even number, and vice versa.  Thus the two sets have the same
"cardinality," and are the same size.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/   

Date: 06/14/2001 at 13:03:37
From: Danielle
Subject: Re: Numbers

Thank you, but I still don't get it.

Date: 06/14/2001 at 22:07:24
From: Doctor Peterson
Subject: Re: Numbers

Hi, Danielle.

This is a very advanced question, really, and it's hard to answer it 
without using big words and big ideas. I'll try to explain it simply, 
but if you don't follow, you'll have to let me know which parts you 
need help with.

The really hard part is that both sets are infinite: they have no end. 
How can you compare infinite sets? Let's start by thinking about how 
we compare finite sets (sets we can actually count).

How do you decide which of these sets is bigger (that is, has more 
members)?

    { 12,52,87 } and { 2,5,7,9,11,13 }

You don't just say, 12 is greater than 2, and 52 is greater than 5, 
and 87 is greater than 7; that really has nothing to do with the size 
of the set itself. What's important is that, after getting that far, 
we run out of numbers in the first set, but we have more in the second 
set to look at. And that's what makes the second set bigger: when we 
match them up, we run out of one before the other. We could show that 
this way:

    12 <-->  2
    52 <-->  5
    87 <-->  7
             9
            11
            13

Now let's say that any two sets are the same size if we can match up 
their members so that every one in the first set is paired with one in 
the second set, and every one in the second set is paired with one in 
the first set. That makes sense, doesn't it?

Now we claim that the set of even numbers and the set of odd numbers 
are the same size. Neither is bigger. Here's my pairing:

    odd     even
    ---     ----
    1   <-->   2
    3   <-->   4
    5   <-->   6
    ...
    101 <--> 102
    ...

and so on forever. I can't list them all, of course, but it's easy to 
see what rule you can follow to decide which even number matches any 
odd number you choose: just add one. And if you name any even number, 
I can tell you which odd number it matches by subtracting one. So by 
our definition, the two sets must be the same size. That answers your 
question.

Now here's where infinite sets start to be weird. I can prove to you 
that the set of even numbers is the same size as the whole set of 
counting numbers! You'd think it would be only half as big, but here's 
my pairing:

    counting    even
    --------    ----
        1   <-->   2
        2   <-->   4
        3   <-->   6
        ...
        101 <--> 202
        ...

Do you see what I've done? Makes you stop and think, doesn't it?

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/