some of the columns), restrict (the process of eliminating some of the rows), union (a way of combining two tables with similar structures), difference (that lists the rows in one table that are not found in the other), intersect (that lists the rows found in both tables), and product (mentioned above, which combines each row of one table with each row of the other). Depending on which other sources you consult, there are a number of other operators – many of which can be defined in terms of those listed above. These include semi-join, outer operators such as outer join and outer union, and various forms of division. Then there are operators to rename columns, and summarizing or aggregating operators, and if you permit relation values as attributes (relation-valued attribute), then operators such as group and ungroup. The SELECT statement in SQL serves to handle all of these except for the group and ungroup operators. The flexibility of relational databases allows programmers to write queries that were not anticipated by the database designers. As a result, relational databases can be used by multiple applications in ways the original designers did not foresee, which is especially important for databases that might be used for a long time (perhaps several decades). This has made the idea and implementation of relational databases very popular with businesses. Database normalization Relations are classified based upon the types of anomalies to which they're vulnerable. A database that is in the first normal form is vulnerable to all types of anomalies, while a database that is in the domain/key normal form has no modification anomalies. Normal forms are hierarchical in nature. That is, the lowest level is the first normal form, and the database cannot meet the requirements for higher level normal forms without first having met all the requirements of the lesser normal forms. Examples Database An idealized, very simple example of a description of some relvars (relation variables) and their attributes: Customer (Customer ID, Tax ID, Name, Address, City, State, Zip, Phone, Email, Sex) Order (Order No, Customer ID, Invoice No, Date Placed, Date Promised, Terms, Status) Order Line (Order No, Order Line No, Product Code, Qty) Invoice (Invoice No, Customer ID, Order No, Date, Status) Invoice Line (Invoice No, Invoice Line No, Product Code, Qty Shipped) Product (Product Code, Product Description) In this design we have six relvars: Customer, Order, Order Line, Invoice, Invoice