Tuple-oriented relational calculus and domain-oriented relational calculus are two different approaches to expressing queries in the relational model of databases.

In tuple-oriented relational calculus, the query specifies a set of tuples (rows) that satisfy a certain condition. The condition is expressed as a formula that contains variables representing the attributes of the tuples. The query results in a set of tuples that satisfy the condition.

In domain-oriented relational calculus, the query specifies a set of values for certain attributes of the tuples. The condition is expressed as a formula that contains variables representing the attributes of the tuples and the values they should take. The query results in a set of tuples that satisfy the condition.

The main difference between the two approaches is the level of abstraction they operate at. Tuple-oriented calculus focuses on the tuples themselves and their attributes, while domain-oriented calculus focuses on the values that the attributes should take. Tuple-oriented calculus is more expressive and flexible, as it can handle more complex queries involving multiple conditions and nested subqueries. Domain-oriented calculus, on the other hand, is more concise and easier to use for simple queries that involve only a few attributes.

Overall, both approaches are useful in different contexts and have their own strengths and weaknesses. The choice of calculus depends on the specific requirements of the query and the preferences of the user.

The tuple-oriented calculus uses a tuple variables i.e., variable whose only permitted values are tuples of that relation. E.g. QUEL

The domain-oriented calculus has domain variables i.e., variables that range over the underlying domains instead of over relation. E.g. ILL, DEDUCE.