Guided Tour:
Conceptual Modelling with Extended Entity-Relationship

The above diagram states that mobile calls are a kind of calls (IsA link between entities); that phone points are partitioned between cell points and landline points (i.e., any phone point is either a cell or a landline point, but not both: they form a covering and disjoint IsA hierarchy); that home points are among phone points, none of them being a cell point (i.e., the home point entity is disjoint from the cell entity). Each call has at least one destination phone point (mandatory participation of cell to origin), while it has exactly one origin phone point. Mobile calls are related to cells through the mOrigin relationship. Finally, the binary relationship mOrigin is a included in the binary relationship origin.

Which are the consequences of the above EER diagram? ICOM automatically produces the following completed diagram:

It is stated that in the above scenario it is necessarily true (green deduction) that any home point is also a landline point, and that each mobile call may have an origin from at most one cell point. In order to understand why this is true, consider the following.
The class of home points is a sub class of all the phone points, and it is disjoint from the class of cell points. Since any phone point is either a cell point or a landline point, then any home point should necessarily be a landline point.
The mOrigin binary relationship is included in the origin binary relationship, i.e., any pair in mOrigin is also among the pairs in origin. Since each call participates exactly once as first argument to the origin relationship, if I take a generic sub class of calls, such as the class of mobile calls, and a sub relationship of the origin relationship, such as mOrigin, then we can conclude that necessarily each mobile call participates at most once as first argument to the mOrigin relationship. Nothing can be concluded about the minimum participation, since the mOrigin relationship may not contain all calls in the Origin relationship.

Let us now add an IsA link to the original diagram, stating that each cell point is among the landline points:

ICOM now concludes the following:

The cell entity is inconsistent, i.e., does not have any instance, since the disjointness constraint in the IsA link states that there is no element in common between cell and landline phone points. The empty set denoted by the cell entity is the only set which can be at the same time disjoint and a subset of another set. Since the phone point entity is formed by the union of the cell and landline entities, and the cell entity is inconsistent, the landline entity becomes equivalent to the phone point entity.
Moreover, since there is no cell point, there is no pair in the mOrigin relationship as well (i.e., it is inconsistent): the diagram states that any second argument of the mOrigin relationship should be of the cell type.
The mobile call entity is not inconsistent, since it may be populated by calls which have no mOrigin at all (this is possible, since there is no mandatory participation constraint).

Let us now add a cardinality constraint, stating that now each mobile call should participate at least once to the mOrigin relationship (i.e., a mandatory participation constraint):

Now, ICOM deduces that the mobile call entity is inconsistent as well:

If we force the landline point entity to be inconsistent as well (by adding an IsA link between landline points and cell points), there will be a cascade effect. The phone point entity becomes inconsistent, since it is the union of two inconsistent classes. Therefore, the home point entity will be inconsistent as well, since it is a sub entity of an inconsistent entity. The destination and origin relationships will be inconsistent, since there is no phone point participating to them. Given the mandatory participation constraint of the call entity to the destination (or origin) relationship, we can conclude that the call entity is necessarily inconsistent as well:

Let us now consider a different example diagram, showing some reasoning involving attributes and domains:

We have a partition of employees between clerks and managers, plus two more sub-entities of employee, namely rich employee and poor employee, disjoint respectively from the clerk and the manager entities. All the sub entities have the salary attribute restricted to a string of length 8, with the exception of the clerk entity which has the salary attribute restricted to be a string of length 5. Is is easy to see that a necessary consequence of the above diagram is that rich employees are managers and that poor employees are clerks:

However, since that salary attribute for clerks and poor employees have incompatible domains, we can not have any poor employee in any legal database, i.e., the poor employee entity is inconsistent.

As the last example, let us introduce a diagram where some entity and some relationship are defined by means of views over other elements of the schema.

The view language is the DLR description logic. The view language includes two syntactic sorts: one for entities and one for relationships. Full boolean operators are allowed for both entities and relationship expressions, plus a selection operator (selecting tuples in a relationship with a specific entity type in some named role argument) and a unary projection operator (projecting a relationship over a named role argument). A generalised projection operator with cardinality restrictions is available as well.
The above diagram introduces NYPD policemen, each one of them legally killed at least one drug dealer, and mafioso people, each one of them illegally killed at least one drug dealer. Moreover, a relation view has been defined as the relation having the "killed" attribute selected to be of the drug dealer type (the view makes use of the selection operator), and an entity view has been defined as the entity obtained by projecting over the killer attribute the union of the legally and illegally kills relationships.

As it is shown in the deduced diagram below, the relation view is a relationship including both the legally and the illegally kills relationships, while the entity view is clearly a super entity of both the NYPD policeman and the mafioso entities:

In the next example, we introduce a UniversalClass, defined as the union of a generic class and its complement, and an EmptyClass, defined as the intersection of a generic class and its complement:

If the UniversalClass is declared as a subclass of the EmptyClass, the whole schema becomes inconsistent, in the sense that it is impossible to assume the existence of any object in the universe, since if it were existing it would contradict the schema: