$R= \{(x,S)\ \epsilon \ U\times P(U)\ | x\ \epsilon \ S\}$. Is $R$ a function of $U$ to $P(U)$? Explain.

Question

Let $U$ be a nonempty set, and let $R$ be the "element of" relation from $U$ to $P(U)$. That is, $R= \{(x,S)\ \epsilon \ U\times P(U)\ | x\ \epsilon \ S\}$. Is $R$ a function of $U$ to $P(U)$? Explain.

So far, the question has asked me to determine what the $dom(R)$ is and what the $range(R)$ is, which I'm pretty sure I found. Any help would be greatly appreciated, thank you!

Answer 1

A function $F$ (a set of ordered pairs) should have the following property:

for all $a$, $b$, $c$, if $(a,b)\in F$ and $(a,c)\in F$, then $b=c$.

Under what assumption can any element of $U$ belong to at most one subset of $U$?

Since $x\in\{x\}$, for every $x\in U$, the relation is a function if and only if $U$ is a singleton. The other case would be $U=\emptyset$, but your $U$ is nonempty from the outset.