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Codd's theorem states that relational algebra and the domain-independent relational calculus queries, two well-known foundational query languages for the relational model, are precisely equivalent in expressive power. That is, a database query can be formulated in one language if and only if it can be expressed in the other.
The relational algebra uses set union, set difference, and Cartesian product from set theory, and adds additional constraints to these operators to create new ones.. For set union and set difference, the two relations involved must be union-compatible—that is, the two relations must have the same set of attributes.
The following proposition says that for any set , the power set of , ordered by inclusion, is a bounded lattice, and hence together with the distributive and complement laws above, show that it is a Boolean algebra.
Another form of composition of relations, which applies to general -place relations for , is the join operation of relational algebra. The usual composition of two binary relations as defined here can be obtained by taking their join, leading to a ternary relation, followed by a projection that removes the middle component.
The standard relational algebra and relational calculus, and the SQL operations based on them, are unable to express directly all desirable operations on hierarchies. The nested set model is a solution to that problem. An alternative solution is the expression of the hierarchy as a parent-child relation. Joe Celko called this the adjacency list ...
For example, this allows us to find each employee and his or her department, but still show departments that have no employees. Below is an example of a right outer join (the OUTER keyword is optional), with the additional result row italicized:
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