Thursday, March 15, 2007

Cardinal Virtues

Riding the ski lifts with my kids has the virtue of being uninterrupted time which opens the door to many fascinating conversations. Barnacle Boy enjoys numbers. He checks out how many chairs are on each lift, and then generally ruminates on interesting properties of that number. There were 173 chairs on the lift we were riding up the other day, which led to...

"Is 173 prime or composite, Mom?"

"Prime, I think."

This conversation quickly led us to think about whether there are more even or odd integers. One theory developed about half-way up the mountain: if the positive and negative odds and evens are paired up, there are equal numbers of each; now throw zero into the mix, which is even. In this way you would conclude that there are more even integers than odd. Given that we have an expert in the house, we consulted Math Man at dinner, who told us that we have Cantor to thank for the solution to this problem of the cardinality or countability of the integers (and other infinite sets of numbers). The set of of even numbers is exactly the same size as the set of odds. The notion is that if you can map one set onto the other, they have to have the same number of members. He then said, there are the same number of even integers as there are integers divisible evenly by 10. This boggles the mind!

The explanation? If you can write a bijective function (that is one that matches each member of set A with a member of set B and vice versa) that connects one set to the other, they have the same number of elements. For example, the set of squares of the integers is not the same size as the set of integers because while you can write a function that takes one member of the set of integers to one and only one member of the set of squares you can't write a function that goes the other way. Each square matches with two possible integers (the positive one and the negative one). The square of 5 is always 25, while the square root of 25 could be either 5 or -5. In Math Man's terms: you can write an injective relationship, but not a surjective one. Hence, the relationship is not bijective! I understand, but I'm still not sure I believe!