Prove equivalence of definitions of statistical distance

Question

Let $P$ and $Q$ be two distributions over a finite set $U$. Given I already proved the following definitions are equilivant: $$ SD(P, Q) = \underset{S⊆U}{max} \ \left\{ \underset{x←P}{Pr} [x ∈ S] − \underset{x←Q}{Pr}[x ∈ S] \right\} $$ $$ SD(P,Q) = \frac{1}2 \sum_{u\in U}|P(u)-Q(u)| $$ I need to prove the following definition: $$ SD(P, Q) = \underset{D}{max}\ \left\{ \underset{x←P}{Pr} [D(x) = 1] − \underset{x←Q}{Pr}[D(x) = 1] \right\} $$ where $D$ is taken over all deterministic algorithms.