# Covering and overlapping constraints

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.

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|

• If all E1 is in E2, and all E2 is in E3, wouldn't that follow the overlapping constraint? E0 will have 25 elements, and if we pick one element that is present in both E1 and E2 (or E2 and E3), it is no longer disjoint, right? – Yash Chowdhary Sep 11 at 13:26
• @YashChowdhary So then "not following" the overlapping constraint implies a disjoint generalization? Either way it seems I had my definition of the overlapping constraint the wrong way around. – NegativeChameleon Sep 11 at 14:10
• Yup, "not following overlapped constraint" implies a disjoint constraint. @NegativeChameleon – Yash Chowdhary Sep 11 at 14:40
• @YashChowdhary Alright, edited. – NegativeChameleon Sep 11 at 14:45