Can someone give a generic definition of any Definite Language in set builder notation?
In set-builder notation, it's just $E\cup \{vw\mid v\in\Sigma^*\text{ and }w\in H\}$.

So, why is this the same thing as "Languages that can be recognized by the last $k$ characters of the string"

One direction is fairly simple. If $L$&nbsp;can be recognized by looking at the last $k$&nbsp;characters, then there must be some set&nbsp;$H$ of length-$k$ strings such that, if the last $k$&nbsp;characters form a string in&nbsp;$H$, you accept the string, and if they form a string not in&nbsp;$H$, you reject. Furthermore, there may be some strings that you accept even though they have fewer than $k$&nbsp;characters: this set is&nbsp;$E$.

The other direction is a little more complex.  Suppose that $L = E\cup\Sigma^*H$ for some finite $E$ and&nbsp;$H$. We need to show that there is some&nbsp;$k$ and sets $X\subseteq\Sigma^{&lt;k}$ and $Y\subseteq \Sigma^k$ such that $L=X \cup\Sigma^*Y$. Note that this is different from the hypothesis that $L=E\cup\Sigma^*H$ because it's more specific: $E$ and&nbsp;$H$ can be any finite sets of strings, whereas $X$&nbsp;contains only strings of length less than&nbsp;$k$ and $Y$&nbsp;contains only strings of length exactly&nbsp;$k$.

The solution is to take $k$ to be whichever is the larger of:


the length of the longest string in&nbsp;$H$;
one plus the length of the longest string in&nbsp;$E$.


(Since both $E$ and&nbsp;$H$ are finite, each has a longest string, or several longest strings of the same length.)

We start by taking $X=E$, which we're allowed to do because every string in&nbsp;$E$ has length strictly less than&nbsp;$k$.  Now, consider some string $h=h_1\dots h_\ell\in H$.  We have $\ell\leq k$ by the choice of&nbsp;$k$.  If $\ell=k$, we're happy: just add $h$ to&nbsp;$Y$.  Now suppose that $\ell&lt;k$. If a string ends with&nbsp;$h$, then either it has length less than&nbsp;$k$ or it has length at least&nbsp;$k$ and its last $k$&nbsp;characters are $a_1\dots a_{k-\ell}h_1\dots h_\ell$ for some $a_1, \dots, a_{k-\ell}\in\Sigma$. Therefore, we add to&nbsp;$X$ all strings of length less than&nbsp;$k$ that end with&nbsp;$h$, and we add to&nbsp;$Y$ all strings of length exactly&nbsp;$k$ that end with&nbsp;$h$. And we repeat this for every $h\in H$.

I've explained how to construct $X$ and&nbsp;$Y$. I won't write out their formal definitions as sets because those definitions are so full of notation that they're not enlightening.