*Multiplicity of Relationships*

Like the binary relationships of the E/R model, a pair of inverse relationships in ODL can be classified as either many-many, many-one in either direction, or one-one. The type declarations for the pair of relationships tells us which.

1. If we have a many-many relationship between classes C and D, then in class C the type of the relationship is Set<D>, and in class D the type is set <C>.

2. If the relationship is many-one from C to D, then the type of the relationship in C is just D, while the type of the relationship in D is Set<C>.

3. If the relationship is many-one from D to C, then the roles of C and D are reversed in (2) above.

4. If the relationship is one-one, then the type of the relationship in C is just D, and in D it is just C.

Note, that as in the E/R model, we allow a many-one or one-one relationship to include the case where for some objects the "one" is in fact "none". For example, a many-one relationship from C to D might have a missing or "null" value of the relationship in some of the C objects. Certainly, since a D object could be linked with any set of C objects, it is also allowable for that set to be empty for some D objects.**Example :** In "Relationships in ODL / Inverse Relationships" Figure (a), we have the declaration of three classes, Movie, Star, and Studio. The first two of these have already been introduced in "Attributes in ODL" Examples (a) and (b). We also discussed the relationship pair stars and starredIn. Since each of their types uses Set, we see that this pair represents a many-many relationship between Star and Movie.

Studio objects have attributes name and address; these appear in lines (13) and (14). Note that the type of addresses here is a string, rather than a structure as was used for the address attribute of class Star on line (10). There is nothing wrong with using attributes of the same name but different types in different classes.

In line (7) we see a relationship ownedBy from movies to studios. Since the type of the relationship is Studio, and not Set<Studio>, we are asserting that for each movie there is one studio that owns it. The inverse of this relationship is found on line (15). There we see the relationship owns from studios to movies. The type of this relationship is Set<Movie>, showing that each studio owns a set of movies - perhaps 0, perhaps 1, or perhaps a large number of movies.

### Tags

- binary relationship
- inverse relationship
- attributes
- e/r model
- Full-Relation Operations
- Natural Joins / Outerjoins
- Subqueries in FROM Clauses
- Conditions Involving Tuples
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- Selection in SQL
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- Additional Constraint Examples
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- A Linear Notation for Algebraic Expressions
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- Selection / Cartesian Product
- Set Operations on Relations
- An Algebra of Relational Operations
- Relational Algebra
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- Representing ODL Relationships
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- Declaring Keys in ODL
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- Types in ODL
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- Relationships in ODL / Inverse Relationships
- Attributes in ODL
- Introduction to ODL
- The Type System
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