=== Base Templates !CardinalityEnd1Min, !CardinalityEnd1Max, !CardinalityEnd1MinMax, !CardinalityEnd2Min, !CardinalityEnd2Max, and !CardinalityEnd2MinMax ===

Are templates for expressing cardinality constraints on relations.
[[br]]
[[br]]!CardinalityEnd1MinMax(a, b) means that a is a relation and b and c are integers, and that
[[br]]the first role of a has b as minimum, c as maximum cardinality. The other templates in this
[[br]]group follow the same pattern.
[[br]]
[[br]]!CardinalityEnd1Min
[[br]]
[[br]]Roles:
[[br]]1 hasRelationship !ClassOfRelationship
[[br]]2 valMinimumCardinality INTEGER
[[br]]
[[br]]!CardinalityEnd1Max
[[br]]
[[br]]Roles:
[[br]]1 hasRelationship !ClassOfRelationship
[[br]]2 valMaximumCardinality INTEGER
[[br]]
[[br]]!CardinalityEnd1MinMax
[[br]]
[[br]]Roles:
[[br]]1 hasRelationship !ClassOfRelationship
[[br]]2 valMinimumCardinality INTEGER
[[br]]3 valMaximumCardinality INTEGER
[[br]]
[[br]]!CardinalityEnd2Min
[[br]]
[[br]]Roles:
[[br]]1 hasRelationship !ClassOfRelationship
[[br]]2 valMinimumCardinality INTEGER
[[br]]
[[br]]!CardinalityEnd2Max
[[br]]
[[br]]Roles:
[[br]]1 hasRelationship !ClassOfRelationship
[[br]]2 valMaximumCardinality INTEGER
[[br]]
[[br]]!CardinalityEnd2MinMax
[[br]]
[[br]]Roles:
[[br]]1 hasRelationship !ClassOfRelationship
[[br]]2 valMinimumCardinality INTEGER
[[br]]3 valMaximumCardinality INTEGER
[[br]]
[[br]]Axiom:
[[br]]
{{{
CardinalityEnd1Min(x1, x2) <->
ClassOfRelationship(x1) &
INTEGER(x2) &
exists u(CardinalityMin(u, x2) & hasEnd1Cardinality(x1, u)) .

}}}
[[br]]Axiom:
[[br]]
{{{
CardinalityEnd1Max(x1, x2) <->
ClassOfRelationship(x1) &
INTEGER(x2) &
exists u(CardinalityMax(u, x2) & hasEnd1Cardinality(x1, u)) .

}}}
[[br]]Axiom:
[[br]]
{{{
CardinalityEnd1MinMax(x1, x2, x3) <->
ClassOfRelationship(x1) &
INTEGER(x2) &
INTEGER(x3) &
exists u(CardinalityMinMax(u, x2, x3) &
hasEnd1Cardinality(x1, u)) .

}}}
[[br]]Axiom:
[[br]]
{{{
CardinalityEnd2Min(x1, x2) <->
ClassOfRelationship(x1) &
INTEGER(x2) &
exists u(CardinalityMin(u, x2)&
hasEnd2Cardinality(x1, u)) .

}}}
[[br]]Axiom:
[[br]]
{{{
CardinalityEnd2Max(x1, x2) <->
ClassOfRelationship(x1) &
INTEGER(x2) &
exists u(CardinalityMax(u, x2)&
hasEnd2Cardinality(x1, u)) .

}}}
[[br]]Axiom:
[[br]]
{{{
CardinalityEnd2MinMax(x1, x2, x3) <->
ClassOfRelationship(x1) &
INTEGER(x2) &
INTEGER(x3) &
exists u(CardinalityMinMax(u, x2, x3)&
hasEnd2Cardinality(x1, u)) .

}}}
[[br]]NOTE For unlimited minimum or maximum cardinalities, assign the values 0 and * cardinality,
[[br]]respectively. See the definitions of templates !CardinalityMin, etc., above.
[[br]]
[[br]]NOTE What constitutes the first and second roles of a relation is stated in the table of prototemplates
[[br]](D.1, page 103).
[[br]]
[[br]]NOTE Let R be a relation to which a cardinality n1 : m1 is assigned to the first role, and a cardinality
[[br]]n2 : m2 to the second. This means that (1) each instance of the first role is R-related to at least
[[br]]n2 and at most m2 distinct instances of the second role, and (2) each instance of the second role is
[[br]]related to at least n1 and at most m1 distinct instances of the first role.
[[br]]
[[br]]EXAMPLE The statement !CardinalityEnd1MinMax(6 M8 bolt assembly, 6, 6) is suitable for expressing
[[br]]an example of a constraint given in ISO 15926-2, clause 4.10.3, that “[t]he
[[br]]class_of_relationship_with_signature ‘6 M8 bolt assembly’ has a cardinality such that each ‘6 of
[[br]]M8 bolts’ is linked by exactly 6 relationships to different M8 bolts at all times”. The expansion of
[[br]]this statement is shown in the following diagram.
[[br]]