Proof Writing & Problem Solving · Superchallenge problem

A relation $R$ on $\{1, 2, 3\}$ is any set of ordered pairs $(x, y)$ with $x, y \in \{1, 2, 3\}$; there are $2^9 = 512$ of them. For how many relations $R$ is the statement "for every $x$ there is a $y$ with $(x, y) \in R$" true, but the statement "there is a $y$ such that for every $x$, $(x, y) \in R$" false?