When I first saw the rationals placed in 1-1 correspondence with the integers I pointed out there were lots of mappings that don't work, e.g., mapping every int

When I first saw the rationals placed in 1-1 correspondence with the integers I pointed out there were lots of mappings that don't work, e.g., mapping every integer to itself, thereby not having any integers left over for nontrivial fractions. The instructor didn't have the right answer handy, and many of us were confused. Sometimes Z and Q had the same cardinality, sometimes they didn't? Here's one answer: it's like saying you can get from New York to Boston, then producing a map that demonstrates the route. There are many other routes that don't take you to Boston, but so what? All you need do is demonstrate A map that gets you there -- you don't care about all the wrong ways to do something. I wish the instructor had hit me over the head with that at the time. Perhaps we were too busy dodging Velociraptors and chasing those cute Neanderthal girls...