2.8: Some special sets

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In addition to the empty set, there are symbols for some other common sets, including:

\(\mathbb{Z}\) — the integers (positive, negative, and zero)
\(\mathbb{N}\) — the natural numbers (positive integers and zero)
\(\mathbb{Q}\) — the rational numbers (all numbers that can be expressed as an integer divided by another integer)
\(\mathbb{R}\) — 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.