2.8: Some special sets
In addition to the empty set, there are symbols for some other common sets, including:
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\(\mathbb{Z}\) — the integers (positive, negative, and zero)
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\(\mathbb{N}\) — the natural numbers (positive integers and zero)
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\(\mathbb{Q}\) — the rational numbers (all numbers that can be expressed as an integer divided by another integer)
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\(\mathbb{R}\) — the real numbers (all numbers that aren’t imaginary, even decimal numbers that aren’t rational)
The cardinality of all these sets is infinity, although as I alluded to previously, \(|\mathbb{R}|\) is in some sense “greater than" \(|\mathbb{N}|\) . For the curious, we say that \(\mathbb{N}\) is a countably infinite set, whereas \(|\mathbb{R}|\) is uncountably infinite . Speaking very loosely, this can be thought of this way: if we start counting up all the natural numbers 0, 1, 2, 3, 4, …, we will never get to the end of them. But at least we can start counting . With the real numbers, we can’t even get off the ground. Where do you begin? Starting with 0 is fine, but then what’s the “next" real number? Choosing anything for your second number inevitably skips a lot in between. Once you’ve digested this, I’ll spring another shocking truth on you: \(|\mathbb{Q}|\) is actually equal to \(|\mathbb{N}|\) , not greater than it as \(|\mathbb{R}|\) is. Cantor came up with an ingenious numbering scheme whereby all the rational numbers — including 3, \(-9\) , \(\frac{4}{17}\) , and \(-\frac{1517}{29}\) — can be listed off regularly, in order, just like the integers can. And so \(|\mathbb{Q}|=|\mathbb{N}|\neq|\mathbb{R}|\) . This kind of stuff can blow your mind.