= Solution
The standard semantics of <ALC> and <EL> do not impose the <unique name assumption>. Therefore the two individual names may denote the same element. A minimum-size model of either normalized <TBox> from part (d), together with the <ABox>, is
$$
\boxed{\begin{gathered}
\Delta^{\mathcal I}=\{d\},\qquad a^{\mathcal I}=b^{\mathcal I}=d,\\
A^{\mathcal I}=C^{\mathcal I}=X^{\mathcal I}=\{d\},\qquad B^{\mathcal I}=\varnothing,\\
R^{\mathcal I}=P^{\mathcal I}=\{(d,d)\}.
\end{gathered}}
$$
The $R$ self-loop supplies the <ABox> fact and makes both $\exists R.A$ and $\exists R.X$ true at $d$. The $P$ self-loop supplies exactly the existential restriction defining $C$. Disjointness holds because $B$ is empty. Hence every normalized <general concept inclusion> is satisfied. A domain must be nonempty, so $\boxed{|\Delta^{\mathcal I}|=1}$ is minimal.
If a separate <unique name assumption> were imposed, the minimum would instead be two. One can take $a\ne b$, $A=\{a\}$, $C=\{b\}$, $R=\{(a,b)\}$, $P=\{(b,b)\}$ and $B=X=\varnothing$. There is then no $R$-edge ending at an $A$ element, so the normalized requirements involving $X$ hold vacuously. That additional convention is not part of standard <description logic> semantics.
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