Name: Brian Provost
Who is asking: Other
Level: All

Here's the deal: There are an infinite amount of integers (1,2,3...). Agreed? There are an infinite amount of even integers, too (2,4,6...). Agreed? By convention, infinity equals infinity. Yet common sense tells us there are obviously more integers than there are even integers. Prove this to be true mathematically.

Hi Brian,

One way to compare the size of two collections of things is to try to match the items in the two collections. For example suppose you are organizing a meeting in some hall and you have put out chairs for the people sit on. Quite a few people have arrived and you wonder if you need to get more chairs. One way to determine this without counting the chairs and people is to ask everyone to sit down. If everyone sits down and there are chairs left then there are more chairs then people. If all the chairs are occupied and some people are left standing there are more people then chairs. If everyone is seated and there are no unoccupied chairs then there are the same number of chairs as people.

This is the concept we use to compare the "size" of sets, whether they are finite or infinite. The word used for "size" is cardinality. Two sets have the same cardinality if there is an exact matching of the items in the first set with the items in the second set. It is precisely this concept that leads to the observation you made that doesn't agree with your "common sense". The positive integers {1, 2, 3, ...} and the even integers {2, 4, 6, ...} have the same cardinality. The matching is

Positive integers Even integers

That is, the number n in the left column is matched with 2n in the right column.

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