Explain what is meant by saying that a binary relation $r \subseteq a \times a$ is well-founded. Show that $r$ is well-founded if and only if, for any set $b$ and any function $f: \mathcal{P} b \rightarrow b$, there exists a unique function $g: a \rightarrow b$ satisfying

$$g(x)=f(\{g(y) \mid\langle y, x\rangle \in r\})$$

for all $x \in a$. [Hint: For 'if', it suffices to take $b=\{0,1\}$, with $f: \mathcal{P} b \rightarrow b$ defined by $\left.f\left(b^{\prime}\right)=1 \Leftrightarrow 1 \in b^{\prime} .\right]$