I'd like to ask for some clarification on basic concepts on relations and functions that I am going over again right now. I would like to present an example I made up.
Let there be four people, Adrian $=A$, Becka $=B$, Carl $=C$ and Dolores $=D$. $A$ and $B$ are grouped into a set we denote $X = \{A,B\}$ and $C$ and $D$ are grouped into a set we denote $Y = \{C,D\}$.
We now seek to express the relation $R=\{(x,y)\in X\times Y: x \in X \text{ is child of } y \in Y\}$.
We furthermore define $R[x]=\{y\in Y: (x,y)\in R\}$.
In our example, it turns out Dolores has been unfaithful and Becka is not the child of Carl. Hence we have
$R=\{(A,C),(A,D),(B,D)\}$.
Am I right in that $R[x]=\{C,D\}$? My textbook claims we can furthermore define a function $f: X \rightarrow \mathcal{P}(Y), x\mapsto R[x]$, and express every relation as this function.
We have $\mathcal{P}(Y)=\{ \emptyset, \{C\},\{D\},\{C,D\} \}$, yes? Can anybody tell me what the image of $f(x)$ would be or if I went wrong anywhere?
you are right about $R[x]={C,D}$.
$f:X\rightarrow Y $is a function if and only if for $$\forall x \in X , \forall y_{1,}y_{2}\in Y ((x,y_{1}), (x,y_{2})\in f \longrightarrow y_{1}=y_{2}))$$ and if $$\forall x\in X ,{\exists} y \in Y$$ so that $$(x, y)\in f$$ i'm not sure what is the function you'd like to define. there's a difference between a function and a relation. if the relation exists this properties, than it's a function.