 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, isThe $R$ 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.