Covering and overlapping constraints

Question

Given that there are 15 entities in E1, 20 entities in E2 and 25 entities in E3, what's the maximum and the minimum number of entities in E0? It's also known that the ISA hierarchy follows the covering constraint but not the overlapping constraint.

I get that the maximum number would be 60, but I'm confused when it comes to the minimum value.

What I'm confused about is if E0 can have 0 entities (or fewer than 15+20+25=60 entities). I understand that unioning all entities in E1, E2, and E3 will give us all entities in E0, but can it be empty too?

Thanks.

Answer 1

The generalization

• follows the covering constraint, so: E1 ⋃ E2 ⋃ E3 = E0

• does not follow the overlapping 'constraint' ⇒ is disjoint, so: E1 ⋂ E2 ⋂ E3 = ∅

The maximum and minimum amount of entities in E₀ is 60. Since no overlapping is allowed there will be 15 + 20 + 25 = 60 distinct entities, which are all entities of superentity E₀ at the same time because of the covering constraint.

E₁ ⋃ E₂ ⋃ E₃ = E₀ ⇒ max{|E₁|, |E₂|, |E₃|} ≤ |E₀| ≤ |E₁| + |E₂| + |E₃|

(E₁ ⋃ E₂ ⋃ E₃ = E₀ ⋀ E₁ ⋂ E₂ ⋂ E₃ = ∅) ⇒ |E₀| = |E₁ + E₂ + E₃|