Q$1,(9$ points$)$ Decide whether the following statement is true or false. If it is true, give a short explanation. If it is false, give a$.$counterexample.

In every instance of the Stable Matching Problem, there is a stable matching containing a pair $(m,w)$ such that $m$ is ranked first on the preference list of $w$ and $w$ is ranked first on the preference list of $m.$

Q$2.(9$ points$)$ Decide whether the following statement is true or false. If it is true, give a short explanation. If it is false, give a counterexample.

Consider an instance of the Stable Matching Problem in which there exists a man $m$ and woman $w$ such that $m$ is ranked first on the preference list of $w$ and $w$ is ranked first on the preference list of $m.$ Then in every stable matching $S$ for this instance, the pair $(m,w)$ belongs to $S.$

$2$