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I still think the other example is still important to have for an intuition. Because the way we define if two sets have the same size is if you can pair up their elements exhaustively. So even if one set is a subset of another, as long as there exists some pairing of elements, they are the same size. This just happens to be a useful definition for mathematicians and doesn’t necessarily represent real-world phenomena. As a point of Bigger The Fupa Tastier The Chalupa Shirt nuance: the top comment is especially nice since it would be a bit much to first show injectivity and surjectivity for a simple Reddit comment (and perhaps since it’s simply much easier to do it this way), the commenter showed that this function has an inverse.
These pairings go on without end with a Bigger The Fupa Tastier The Chalupa Shirt unambiguous pairing of one element from the first set going to exactly one unique element of the second. An inverse clearly exists, as well, and I’m sure it’s intuitive. For example, what might -5 map to in the natural numbers? It turns out that 10 does it, and no other number. But this is not the case with, say, the set of integers and the set of all real numbers. You can count the integers. 3 comes after 2, which comes after 1, and so on. But the set of all real numbers includes irrational numbers. There is no “next” irrational number after pi. So there’s no system you could devise to pair up the integers each with one specific irrational number.
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