Projection (relational algebra)

In relational algebra, a projection is a unary operation written as $\pi _{a_{1},...,a_{n}}(R)$ where $a_{1},...,a_{n}$ is a set of attribute names. The result of such projection is defined as the set that is obtained when all tuples in $R$ are restricted to the set $\{a_{1},...,a_{n}\}$. For an example, consider the following two tables which are the relation $Person$ and its projection on the attributes $Age$ and $Weight$:

$Person$	$\pi _{Age,Weight}(Person)$
Name Age Weight Harry 34 80 Sally 28 64 George 29 70 Helena 54 54 Peter 34 80	Age Weight 34 80 28 64 29 70 54 54

Note that Harry and Peter have the same age and weight, but since the result is a relation, and therefore a set, this combination only appears once in the result.

More formally the semantics of the projection is defined as follows:

$\pi _{a_{1},...,a_{n}}(R)=\{\ t[a_{1},...,a_{n}]:\ t\in R\ \}$

where $t[a_{1},...,a_{n}]$ is the restriction of the tuple $t$ to the set $\{a_{1},...,a_{n}\}$ so that

$t[a_{1},...,a_{n}]=\{\ (a',v)\ |\ (a',v)\in t,\ a'\in a_{1},...,a_{n}\ \}$

The result of a projection $\pi _{a_{1},...,a_{n}}(R)$ is only defined if $\{a_{1},...,a_{n}\}$ is a subset of the header of $R$.

In SQL, projections are done by using the SELECT statement.