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In practical terms, if a relation is thought of as a table, then projection can be thought of as picking a subset of its columns. For example, if the attributes are (name, age), then projection of the relation {(Alice, 5), (Bob, 8)} onto attribute list (age) yields {5,8} – we have discarded the names, and only know what ages are present.
In relational algebra, a selection (sometimes called a restriction in reference to E.F. Codd's 1970 paper [1] and not, contrary to a popular belief, to avoid confusion with SQL's use of SELECT, since Codd's article predates the existence of SQL) is a unary operation that denotes a subset of a relation.
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.
For example, the mapping that takes a point (x, y, z) in three dimensions to the point (x, y, 0) is a projection. This type of projection naturally generalizes to any number of dimensions n for the domain and k ≤ n for the codomain of the mapping. See Orthogonal projection, Projection (linear algebra). In the case of orthogonal projections ...
A relation that is functional and total. For example, the red and green relations in the diagram are functions, but the blue and black ones are not. An injection [d] A function that is injective. For example, the green relation in the diagram is an injection, but the red, blue and black ones are not. A surjection [d] A function that is surjective.
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The table T above is an example of Codd-table. Codd-table algebra supports projection and positive selections only. It is also demonstrated in [IL84 that it is not possible to correctly extend more relational operators over Codd-Tables. For example, such basic operation as join is not extendable over Codd-tables.
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