<p><pre><code> Bijection is a simple idea. But it’s an important tool because it helps keep us
out of the counterintuitive weeds when working with infinite sets. For instance,
we can now figure this out: are there more positive integers {1, 2, 3, ...} or
even integers {2, 4, 6, ...}? The naive answer would be that there must be more
positive integers, because the set of positive integers includes both the even
and odd integers.
But this is wrong. Using bijection, we see that we can put the positive integers
and even integers into a one-to-one correspondence like so:
1, 2, 3, 4, ...
2, 4, 6, 8, ...
</code></pre>
Couldn't you also reason that there are more even integers than positive integers if you start with the notion that you start with a number and start counting in the direction(s) of the defined set:<p><pre><code> X--->
1, 2, 3, 4, ...
..., -4, -2, 0, 2, 4, 6, 8, ...
<--X-->
</code></pre>
The set of all integers progresses along two vectors with each "count" and only positive increases along a single vector with each count.