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*.

- “
*Draw*” gray $A$ and white $B$. - “
*Draw*” the given injections $f: A \to B$ ($A$ contained in $B$) and $g: B \to A$ ($B$ contained in $A$). (Figure 13.4) - “
*Draw*” an*infinite chain*of*alternating*injections starting from $A$. (Figure 13.5) - In “
*diagram $A$ at step infinity $\infty$*” ($A$ containing $B$ containing $A$ …), label the “*gray region*” as $G$. - Label remaining white region as $W$. (i.e. $W := A \setminus G$)
- “
*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) - 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 $f(A)$.

- Applying $f$ on $W$

It’s nice to see a constructive and *formal* proof immediately following this
intriguing argument. The later actually guides me through the former.