Understanding the Choice Rule in MWG
I am having trouble understanding what $C(\{x, y, z\}) = \{x, y\}$ means.

MWG already explain this on p.10:

When [$C(B)$ contains more than one element], the elements of $C(B)$ are the alternatives in $B$ that the decision maker might choose; that is, they are her acceptable alternatives in $B$. In this case, the set $C(B)$  can be thought of as containing those alternatives that we would actually see chosen if the decision maker were repeatedly to face the problem of choosing an alternative from set $B$.



[H]ow can we even have a situation where if the consumer is choosing between $x$ and $y$, he'll always choose $x$, but then when he is faced between choosing $x$, $y$, and $z$, he will choose either $x$ or $y$? Can someone give me a real life example of how this would be possible?

Consider a voting situation as follows. Individuals $x$, $y$, and $z$ are potential candidates for the position of a department chair. A five-member committee is to select the chair by voting via simple majority rule. Let's label the committee members $1$ to $5$ and suppose that they each have a private ranking of the three candidates as follows ($\succ_i$ is member $i$'s ranking):
\begin{align}
x\succ_1 y\succ_1 z \\
y\succ_2 x\succ_2 z \\
z\succ_3 x\succ_3 y \\
y\succ_4 z\succ_4 x \\
x\succ_5 z\succ_5 y 
\end{align}
Suppose that each member will vote for his/her top-ranked candidate. Then when only two candidates $x$ and $y$ are under consideration, $x$ will get three votes (from members $1$, $3$ and $5$) and thus always be chosen. But when all three candidates are being considered, $x$ and $y$ will each get two votes ($x$ from $1$ and $5$, $y$ from $2$ and $4$) and thus both be &quot;acceptable&quot; candidates.