= Solution
Take distinct classes $p(x)\neq p(y)$. The points are not related by $R$, so by the defining negation there is a <Hausdorff space> $Z$ and a <continuous map> $f:X\to Z$ with $f(x)\neq f(y)$. Choose disjoint open neighborhoods $U$ and $V$ of those two values. Their inverse images $A=f^{-1}(U)$ and $C=f^{-1}(V)$ are disjoint open subsets of $X$ and are saturated by part (ii).
Hence $p^{-1}(p(A))=A$ and $p^{-1}(p(C))=C$, making $p(A)$ and $p(C)$ open in the <quotient topology>. They contain the chosen classes and are disjoint: an <equivalence class> meeting both would contradict saturation and $A\cap C=\varnothing$. Thus \b[$X/R$ is Hausdorff]. This quotient is the <Hausdorff reflection by continuous maps>; every <continuous map> from $X$ to a <Hausdorff space> factors uniquely through it.
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