$L_{\omega_1}$ is the $\omega_1$-th constructible hierarchy. I define two binary relation on $P(\omega_1)$ as follows:

For $X,Y\in{P}(\omega_1)$,

$R_1(X,Y)$ means: there is $U\subset\omega_1\times\omega_1$ which is $\Sigma_1$ definable with parameters in $L_{\omega_1}$ such that $Y=\lbrace x\in\omega_1\mid\exists{t}\in{X}(t,x)\in{U}\rbrace$.

$R_2(X,Y)$ means: $Y$ is $\Sigma_1(X)$ in the model $(L_{\omega_1},\in)$, i.e. $Y$ is $\Sigma_1$-definable with $X$ as a unary predicate.

The question is:

$A$ is a subset of $\omega_1$, define $A'$ as follows: $A'$ is the set of all pairs $(a,b)$, $a,b\in\omega_1$, $K(a)\subset{A}$ and $K(b)\subset\omega_1\setminus{A}$.

($K:\omega_1\rightarrow{L_{\omega_1}}$ is the canonical emumeration of elements of $L_{\omega_1}$.)

Then is $R_1(A,A')$ true?

Are $R_1$ and $R_2$ equivalent?

Is $R_1$ or $R_2$ transitive?

`$L_{\omega_1}$`

, then this situation is familiar from recursion theory: $R_1(X,Y)$ then is a weak form of enumeration reducibility of $Y$ to $X$, while $R_2(X,Y)$ says that $Y$ is r.e. relative to $X$. These relations are distinct, and $R_2$ is not transitive --- because of complementation issues as in Joel's answer. – Andreas Blass Aug 19 '12 at 14:40