Proving that the two definitions of Universal Class of Hash Function are equivalent (as dealt with in CLRS)
Both definitions are equivalent, and I think you have almost specified the reason behind it.
The below equivalence work in both ways.
From Definition 2:
&quot;the chance of a collision between distinct keys $k$ and $l$&quot; $\le$ $1/m$
$\Leftrightarrow$
{note that the &quot;chance&quot; here is about the process of picking $h$ from the class $\mathscr{H}$}
(Number of functions in the class where $k$ and $l$ collide) $/$ (Total functions in the class) $\le$ $1/m$
$|A|/|\mathscr{H}| \le 1/m$
$\Leftrightarrow$
$|A| \le |\mathscr{H}|/m$
(which is the wording used in Definition 1)