Apart from the standard definition of a relation, Recently read this one in a book:
Let E and F be two sets.
A relation $τ$ can also be viewed as a function from $E$ to $P(F)$ by setting,
for each $x ∈ E$,
$τ (x) = \{{y ∈ F | (x, y) ∈ τ\}}$
I am not able to understand this recursive definition.
This definition is not recursive, it is just badly notated.
If $\tau \subseteq E \times F$ is a relation, define a function $f_\tau : E \to 2^F$ by
$f_\tau(x) = \{ y ~|~ (x,y) \in \tau \}$
Here $2^F$, as usual, denotes the powerset of $F$
Edit: here's a small example:
Say $E = \{ a,b,c \}$, $F = \{ x,y \}$, $\tau = \{ (a,x), (a,y), (b,x), (c,y) \}$.
Then: $$f_\tau(a) = \{ x,y \}$$ $$f_\tau(b) = \{ x \}$$ $$f_\tau(c) = \{ y \}$$
Hope this helps! ^_^