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No, because with infinities, even if a set is a subset of another, you may still be able to find one element in the first set that corresponds to every element of the other set. For example, the natural numbers are a subset of the whole numbers, but there is a natural number that corresponds to every whole number. To see this, we can order the whole numbers like this: {0, -1, 1, -2, 2, -3, 3, ...}, and we can easily see that we can now assign one natural number to each of them (0 -> 0, -1 -> 1, 1 -> 2, -2 -> 3, ...). Since you'll never run out of naturals, you won't ever find a whole number that doesn't have a corresponding natural number. Since assigning a natural number this way is equivalent to counting the elements of the other set (in this scheme, I could say that -2 is the 3rd whole number), this type of infinity is called "countable infinity". The natural numbers, the whole numbers, and the rational numbers are all countably infinite. In contrast, the irrational numbers and the real numbers are not. In fact, even the real interval [0, 1] is not countable, so this interval is considered to have more elements than N (the set of natural numbers). Note that while there is only one countable infinity, there are many uncountable infinities - so not all uncountably infinite sets are considered as large. If you're curious about this area, the study of these concepts is done via "transfinite numbers" - particularly, the transfinite cardinal numbers (there are also transfinite ordinals). |