Formal definitions

A Relation may be defined in multiple ways. The Schema of a Relation: R (A1, A2, .....An) Relation schema R is defined over attributes A1, A2, .....An.
For Example -
CUSTOMER (Cust-id, Cust-name, Address, Phone#)
Here, CUSTOMER is a relation defined over the four attributes Cust-id, Cust-name, Address, Phone#, each of which has a domain or a set of valid values. For example, the domain of Cust-id is 6 digit numbers.
A tuple is an ordered set of values.Each value is derived from an appropriate domain. Each row in the CUSTOMER table may be referred to as a tuple in the table and would consist of four values.
<632895, "John Smith", "101 Main St. Atlanta, GA 30332", "(404) 894-2000">
is a tuple belonging to the CUSTOMER relation.
A relation may be regarded as a set of tuples (rows). Columns in a table are also called attributes of the relation.
A domain has a logical definition: e.g.,
“USA_phone_numbers” are the set of 10 digit phone numbers valid in the U.S. A domain may have a data-type or a format defined for it. The USA_phone_numbers may have a format: (ddd)-ddd-dddd where each d is a decimal digit. E.g., Dates have various formats such as monthname, date, year or yyyy-mm-dd, or dd mm,yyyy etc.
An attribute designates the role played by the domain. E.g., the domain Date may be used to define attributes “Invoice-date” and “Payment-date”.
The relation is formed over the cartesian product of the sets; each set has values from a domain; that domain is used in a specific role which is conveyed by the attribute name.
For example, attribute Cust-name is defined over the domain of strings of 25 characters. The role these strings play in the CUSTOMER relation is that of the name of customers.
Given R(A1, A2, .........., An)
r(R) Ì dom (A1) X dom (A2) X ....X dom(An)
R: schema of the relation
r of R: a specific "value" or population of R.
R is also called the intension of a relation
r is also called the extension
Let S1 = {0,1}
Let S2 = {a,b,c}
Let R Ì S1 X S2
Then for example: r(R) = {<0,a> , <0,b> , <1,c> } is one possible “state” or

“population” or “extension” r of the relatio R, defined over domains S1 and S2. It has three tuples.