Join (relational algebra)

θ-join and equijoin

Consider tables Car and Boat which list models of cars and boats and their respective prices. Suppose a customer wants to buy a car and a boat, but she does not want to spend more money for the boat than for the car. The θ-join (⋈_θ) on the predicate CarPrice ≥ BoatPrice produces the flattened pairs of rows which satisfy the predicate. When using a condition where the attributes are equal, for example Price, then the condition may be specified as Price=Price or alternatively (Price) itself.

In order to combine tuples from two relations where the combination condition is not simply the equality of shared attributes it is convenient to have a more general form of join operator, which is the θ-join (or theta-join). The θ-join is a binary operator that is written as ${\displaystyle \genfrac{}{}{0ex}{}{R\bowtie S}{a\theta b}}$ or ${\displaystyle \genfrac{}{}{0ex}{}{R\bowtie S}{a\theta v}}$ where a and b are attribute names, θ is a binary relational operator in the set {<, ≤, =, ≠, >, ≥}, v is a value constant, and R and S are relations. The result of this operation consists of all combinations of tuples in R and S that satisfy θ. The result of the θ-join is defined only if the headers of S and R are disjoint, that is, do not contain a common attribute.

The simulation of this operation in the fundamental operations is therefore as follows:

R ⋈_θ S = σ_θ(R × S)

In case the operator θ is the equality operator (=) then this join is also called an equijoin.

Note, however, that a computer language that supports the natural join and selection operators does not need θ-join as well, as this can be achieved by selection from the result of a natural join (which degenerates to Cartesian product when there are no shared attributes).

In SQL implementations, joining on a predicate is usually called an inner join, and the on keyword allows one to specify the predicate used to filter the rows. Forming the flattened Cartesian product then filtering the rows is conceptually correct, but an implementation would use more sophisticated data structures to speed up the join query.

Semijoin

The left semijoin (⋉ and ⋊) is a joining similar to the natural join and written as ${\displaystyle R⋉S}$ where ${\displaystyle R}$ and ${\displaystyle S}$ are relations. The result is the set of all tuples in ${\displaystyle R}$ for which there is a tuple in ${\displaystyle S}$ that is equal on their common attribute names. The difference from a natural join is that other columns of ${\displaystyle S}$ do not appear. For example, consider the tables Employee and Dept and their semijoin:

More formally the semantics of the semijoin can be defined as follows:

${\displaystyle R⋉S=\{t:t\in R\wedge \mathrm{\exists }s\in S(\mathrm{Fun}(t\cup s))\}}$

where ${\displaystyle \mathrm{Fun}(r)}$ is as in the definition of natural join.

The semijoin can be simulated using the natural join as follows. If ${\displaystyle a_1,\dots ,a_n}$ are the attribute names of ${\displaystyle R}$, then

${\displaystyle R⋉S={\mathrm{\Pi }}_{a_1,\dots ,a_n}(R\bowtie S).}$

Since we can simulate the natural join with the basic operators it follows that this also holds for the semijoin.

In Codd's 1970 paper, semijoin is called restriction.

Antijoin

The antijoin (▷), written as R ▷ S where R and S are relations, is similar to the semijoin, but the result of an antijoin is only those tuples in R for which there is no tuple in S that is equal on their common attribute names.

For an example consider the tables Employee and Dept and their antijoin:

The antijoin is formally defined as follows:

R ▷ S = { t : t ∈ R ∧ ¬∃s ∈ S(Fun (t ∪ s))}

or

R ▷ S = { t : t ∈ R, there is no tuple s of S that satisfies Fun (t ∪ s)}

where Fun (t ∪ s) is as in the definition of natural join.

The antijoin can also be defined as the complement of the semijoin, as follows:

Given this, the antijoin is sometimes called the anti-semijoin, and the antijoin operator is sometimes written as semijoin symbol with a bar above it, instead of ▷.

In the case where the relations have the same attributes (union-compatible), antijoin is the same as minus.

Division

The division (÷) is a binary operation that is written as R ÷ S. Division is not implemented directly in SQL. The result consists of the restrictions of tuples in R to the attribute names unique to R, i.e., in the header of R but not in the header of S, for which it holds that all their combinations with tuples in S are present in R.

Example

If DBProject contains all the tasks of the Database project, then the result of the division above contains exactly the students who have completed both of the tasks in the Database project. More formally the semantics of the division is defined as follows:

where {a₁,...,a_n} is the set of attribute names unique to R and t[a₁,...,a_n] is the restriction of t to this set. It is usually required that the attribute names in the header of S are a subset of those of R because otherwise the result of the operation will always be empty.

The simulation of the division with the basic operations is as follows. We assume that a₁,...,a_n are the attribute names unique to R and b₁,...,b_m are the attribute names of S. In the first step we project R on its unique attribute names and construct all combinations with tuples in S:

T := π_a1,...,an(R) × S

In the prior example, T would represent a table such that every Student (because Student is the unique key / attribute of the Completed table) is combined with every given Task. So Eugene, for instance, would have two rows, Eugene → Database1 and Eugene → Database2 in T.

EG: First, let's pretend that "Completed" has a third attribute called "grade". It's unwanted baggage here, so we must project it off always. In fact in this step we can drop "Task" from R as well; the multiply puts it back on.

T := π_Student(R) × S // This gives us every possible desired combination, including those that don't actually exist in R, and excluding others (eg Fred | compiler1, which is not a desired combination)

In the next step we subtract R from T

relation:

U := T − R

In U we have the possible combinations that "could have" been in R, but weren't.

EG: Again with projections — T and R need to have identical attribute names/headers.

U := T − π_Student,Task(R) // This gives us a "what's missing" list.

So if we now take the projection on the attribute names unique to R

then we have the restrictions of the tuples in R for which not all combinations with tuples in S were present in R:

V := π_a1,...,an(U)

EG: Project U down to just the attribute(s) in question (Student)

V := π_Student(U)

So what remains to be done is take the projection of R on its unique attribute names and subtract those in V:

W := π_a1,...,an(R) − V

EG: W := π_Student(R) − V.