Relational Algebra¶
defintion: In the relational model, operations can be expressed in algebra called relational algebra. In relational algebra, relations are the operands and we apply operators to them.
Basic Operations¶
Union¶
Union ($\cup$) is all tuples from each operand. For example,
is all the tuples in $R$ or in $S$, or both.
Intersection¶
Intersection ($\cap$) is all tuples from both operands. For example,
is all the tuples in both $R$ and in $S$.
Difference¶
Difference ($-$) is the difference between the tuples from each operand. For example,
is all the tuples in $R$ but not in $S$. Note $R - S \neq S - R$.
Removal Operations¶
Selection¶
Selection ($\sigma$) eliminates some rows, for example
is a new relation with rows from $R$ where $A$ is less than $2$.
Projection¶
Projection ($\pi$) eliminates some columns, for example
is a new relation with columns $A_1,\ldots,A_n$ from $R$.
Combination Operations¶
Cartesian Product¶
The Cartesian Product ($\times$) is all tuples with the first part from the first operand and the second part from the second operand. Common attributes should be prefixed with the relation name. For example, given the relation $R$
| $A$ | $B$ |
|---|---|
| $a_1$ | $b_1$ |
| $a_2$ | $b_2$ |
and the relation $S$
| $B$ | $C$ | $D$ |
|---|---|---|
| $b_3$ | $c_1$ | $d_1$ |
| $b_4$ | $c_2$ | $d_2$ |
| $b_5$ | $c_3$ | $d_3$ |
then the relation $R \times S$ is
| $A$ | $R.B$ | $S.B$ | $C$ | $D$ |
|---|---|---|---|---|
| $a_1$ | $b_1$ | $b_3$ | $c_1$ | $d_1$ |
| $a_1$ | $b_1$ | $b_4$ | $c_2$ | $d_2$ |
| $a_1$ | $b_1$ | $b_5$ | $c_3$ | $d_3$ |
| $a_2$ | $b_2$ | $b_3$ | $c_1$ | $d_1$ |
| $a_2$ | $b_2$ | $b_4$ | $c_2$ | $d_2$ |
| $a_2$ | $b_2$ | $b_5$ | $c_3$ | $d_3$ |