Uncovered set versus top cycle set (voting theory)
This example comes from Miller (1980) (the paper that introduced the uncovered set definition).

Suppose that the majority's preference is defined by:

\begin{aligned}
x \succ y \\
x \succ z \\
y \succ v \\
y \succ z \\
v \succ x \\
z \succ v
\end{aligned}

In this example, we have the top cycle set $L_1 = \{x, y, z, v \}$ (you can check this).

But notice that $y$ is preferred to $z$ and all policies that are preferred to $y$ (namely $x$), are preferred to $z$. So, $y$ covers $z$ and the uncovered set $U$ is such that $U = \{x, y, v\}$ and $U \subset L_1$.

Among other stuff that Miller proves in his paper, we have the following three:


if there is a Condorcet winner, $L_1 = U$;
if $L_1$ has three elements, then $L_1 = U$;
$U \subseteq L_1$.