[mcguire]

You don't actually walk the sets to compare their sizes. You establish a 1-1 relationship between their elements.

Take the natural numbers 0,1,2,3... (call that N) and the even counting numbers 0,2,4,6... (call that E).

Here's a function between N -> E: f(n) = n * 2. You can prove that f maps every element of N to an element of E, and that every element of E is mappable from an element of N. f is 1-1. And therefore, N and E are the same "size".