Chapter 1.6, Problem 1E

(a) Make a list of all the injective maps

$f:\left\{1,2,3\right\}\to \left\{1,2,3,4\right\}$.

Show that none is bijective. (This constitutes a direct proof that a set $A$ of cardinality three does not have cardinality four.)

(b) How many injective maps

$f:\left\{1,\mathrm{...},8\right\}\to \left\{1,\mathrm{...},10\right\}$

are there? (You can see why one would not wish to try to prove directly that there is no bijective correspondence between these sets.)