For example, is the set of even integers {..., -4, -2, 0, 2, 4, ...} smaller than the set of all integers {...,-4,-3,-2,-1,0,1,2,3,4,....} ?

Our intuition says that removing an element from a set makes the set smaller, so removing an infinite number of elements should make it a lot smaller. That intuition is wrong though.

We say that two sets have the same cardinality if we can make a 1-to-1 correspondence between them. A 1-to-1 correspondance between A and B is a pairing of elements of A and B where every element of A is in exactly one pair, and every element B is in exactly one pair.

Here's a 1-to-1 correspondence between the integers and the even integers: k pairs with 2k. Every integer is on the left side (the k's) and every even integer is on the right side (the 2k's).

## No comments:

## Post a Comment