In description logics, how do you define a concept?

Question

In some texts I read I see a concept defined as a subset and in other texts I see it defined with equivalence.

Eg 1: Person is a subset of hasMother.Person and hasFather.Person

Eg 2: Mother is equivalent to Woman and hasChild.Person

When is it appropriate to use subset and when is it appropriate to use equivalence? I can't seem to find a distinction between the two.

Answer 1

The second one is a defintion; so, it is appropriate to use equivalence.

The concept $\text {Mother}$ is defined as the intersection of the two concepts : $\text {Woman}$ and $\text {hasChild.Person}$.

Reagrding the first one, we have that $\text{Person}$ is a concept, whose meaning is the class of all humans. Also $\text {hasMother.Person}$ is a concept and its meaning is again the class of all humans, because every human has a mother.

Thus the two concepts have the same meaning, and this implies that, given an interpretation $\mathfrak I$ with domain $\Delta$ containing all human beings :

$(\text{Person})^{\mathfrak I} = (\text {hasMother.Person})^{\mathfrak I} \subseteq \Delta$.

In this sense, we have both :

$(\text{Person})^{\mathfrak I} \subseteq (\text {hasMother.Person})^{\mathfrak I} \text { and } (\text {hasMother.Person})^{\mathfrak I} \subseteq (\text{Person})^{\mathfrak I}$.