## Definition of n-ary Relation

### Subjects to be Learned

• ordered n-tuple (review)
• Cartesian product (review)
• equality of n-tuples (review)
• n-ary relation
• empty relation
• universal relation

### Contents

Here we are going to formally define general n-ary relation using the concept of ordered n-tuple.

Definition (ordered n-tuple): An ordered n-tuple is a set of n objects with an order associated with them. If n objects are represented by x1, x2, ..., xn, then we write the ordered n-tuple as <x1, x2, ..., xn> .

Definition (Cartesian product): Let A1, ..., An be n sets. Then the set of all ordered n-tuples <x1, ..., xn> , where xi Ai for all i, 1 i n , is called the Cartesian product of A1, ..., An, and is denoted by A1 ... An .

Definition (equality of n-tuples): Two ordered n-tuples <x1, ..., xn> and <y1, ..., yn> are equal if and only if xi = yi for all i, 1 i n .
For example the ordered 3-tuple <1, 2, 3> can be equal to only <1, 2, 3> and nothing else. It is not equal to the ordered n-tuple <2, 3, 1> for example.

Definition (n-ary relation): An n-ary relation on sets A1, ..., An is a set of ordered n-tuples <a1, ..., an> where ai is an element of Ai for all i, 1 i n . Thus an n-ary relation on sets A1, ..., An is a subset of Cartesian product A1 ... An .

Example 1: Let A1 be a set of names, A2 a set of addresses, and A3 a set of telephone numbers. Then a set of 3-tuples < name, address, telephone number > such as { < Amy Angels, 35 Mediterranean Ave, 224-1357 >, < Barbara Braves, 221 Atlantic Ave, 301-1734 >, < Charles Cubs, 312 Baltic Ave, 223-9876 > }, is a 3-ary (ternary) relation over A1, A2 and A3.

Example 2: Let A1 be a set of names. Then a set of 1-tuples such as { < Amy >, < Barbara >, < Charles > }, is a 1-ary (unary) relation over A1.
A unary relation represents a property/characteristic, such as tall, rich etc., shared by the members of A1 listed in the relation.

#### Special Relations

The empty set is certainly a set of ordered n-tuples. Therefore it is a relation. It is called the empty relation.
The Cartesian product A1 ... An of sets A1 , ... , An ,  is also a relation, and it is called the universal relation.

### Test Your Understanding of n-ary Relation

Indicate which of the following statements are correct and which are not.
Click True or False , then Submit. There is one set of questions.

Next -- Equality of Relations

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