<iclass="fa fa-info-circle"aria-hidden></i> This _isn't_ a substitute for

books <iclass="fa fa-book"aria-hidden></i>.

### 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_][1], 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_][2].

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.