Set Theory · Equinumerosity
Lesson 4
Problem. Test your intuition! Which of the following pairs of sets have the same cardinality (that is, there exists a bijection between their elements)?
\(\mathbb{Z}\) (integers) versus \(\mathbb{Q}\) (rational numbers). It is clear that there are more rational numbers (that is, \(\mathbb{Z}\subset \mathbb{Q}\)), but perhaps there is still a bijection between them?
\(\mathbb{Z}\) (integers) versus \(\mathbb{R}\) (real numbers). Yes, \(\mathbb{Z}\subseteq \mathbb{R}\), but we can still write each real number as a sequence of digits. Maybe that's why the bijection between these sets exists?
\(I=\{x \in \mathbb{R}\colon 0 \le x \le 1\}\) (unit segment) versus \(S=\{(x,y) \in \mathbb{R}^{2} \colon 0 \le x,y \le 1\}\) (unit square). Both of these sets are infinite, but it seems that the square is “more infinite”: its “volume” is non-zero and it seems to be composed of an infinite number of segments (“infinity squared”?).