### Post: CSB Theorem

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 --- title: "CSB Theorem" subtitle: "A visual argument for CSB Theorem" date: 2018-08-29T11:25:24+02:00 categories: - math tags: - set theory --- This _isn't_ a substitute for books . ### Reminder - \$A \preceq B\$: \$A\$ can be "_injected_" into \$B\$ - \$A \sim B\$: \$A\$ and \$B\$ share the _same cardinality_ - \$A \prec B\$: \$A\$ can be "_injected_" into \$B\$, but it's "_smaller_" than \$B\$. - A **finite** set can be "_counted_" from one to some nonnegative integer. - **Infinite** is the "_antonym_" of finite. ### Wolf's proof When I first saw this proof in Robert S. Wolf's [_Proof, Logic and Conjecture: The Mathematician's Toolbox_], I gave it up since it _wasn't_ as intuitive as the statement. ### An informal argument Later, I found some interesting _illustrations_ in Richard Hammack's [_Book of Proof_]. 1. "_Draw_" gray \$A\$ and white \$B\$. 2. "_Draw_" the given injections \$f: A \to B\$ (\$A\$ contained in \$B\$) and \$g: B \to A\$ (\$B\$ contained in \$A\$). (Figure 13.4) 3. "_Draw_" an _infinite chain_ of _alternating_ injections starting from \$A\$. (Figure 13.5) 4. In "_diagram \$A\$ at step infinity \$\infty\$_" (\$A\$ containing \$B\$ containing \$A\$ …), label the "_gray region_" as \$G\$. 5. Label remaining white region as \$W\$. (i.e. \$W := A \setminus G\$) 6. "_Draw_" a "_homologous_" diagram with the one in step 4 on the right-hand side, but starting from \$B\$. (i.e. \$B\$ containing \$A\$ containing \$B\$ …) (Figure 13.6) 7. It's natural to associate the gray regions \$G \subseteq A\$ with \$f(G) \subseteq B\$ on both sides. It remains to settle \$W\$. - Applying \$f\$ on \$W\$ _won't_ lead to any useful results. - Another given injection \$g\$ _can't_ be applied on \$W\$ due to domain mismatch. - Reverse the "_direction_" of \$g\$ to that it points to the white region wrapping gray \$f(A)\$. It's nice to see a constructive and _formal_ proof immediately following this intriguing argument. The later actually guides me through the former. : http://www.columbia.edu/~vml2113/Teachers%20College,%20Columbia%20University/Academic%20Year%202011-2012/Spring%202012/MSTM%206051%20-%20Advanced%20Topics%20in%20Nature%20of%20Proofs/Proof,%20Logic,%20and%20Conjecture%20-%20The%20Mathematician%27s%20Toolbox.pdf : https://www.people.vcu.edu/~rhammack/BookOfProof/BookOfProof.pdf
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