Set operators

The relational algebra uses set union, set difference, and Cartesian product from set theory, but adds additional constraints to these operators.

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. Because set intersection is defined in terms of set union and set difference, the two relations involved in set intersection must also be union-compatible.

For the Cartesian product to be defined, the two relations involved must have disjoint headers—that is, they must not have a common attribute name.

In addition, the Cartesian product is defined differently from the one in set theory in the sense that tuples are considered to be "shallow" for the purposes of the operation. That is, the Cartesian product of a set of n-tuples with a set of m-tuples yields a set of "flattened" (n + m)-tuples (whereas basic set theory would have prescribed a set of 2-tuples, each containing an n-tuple and an m-tuple). More formally, R × S is defined as follows:

$${\displaystyle R\times S:=\{(r_{1},r_{2},\dots ,r_{n},s_{1},s_{2},\dots ,s_{m})|(r_{1},r_{2},\dots ,r_{n})\in R,(s_{1},s_{2},\dots ,s_{m})\in S\}}$$

The cardinality of the Cartesian product is the product of the cardinalities of its factors, that is, |R × S| = |R| × |S|.

Projection (Π)

A projection is a unary operation written as ${\displaystyle \Pi _{a_{1},\ldots ,a_{n}}(R)}$ where ${\displaystyle a_{1},\ldots ,a_{n}}$ 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 ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$.

Note: when implemented in SQL standard the "default projection" returns a multiset instead of a set, and the Π projection to eliminate duplicate data is obtained by the addition of the DISTINCT keyword.

Selection (σ)

A generalized selection is a unary operation written as ${\displaystyle \sigma _{\varphi }(R)}$ where φ is a propositional formula that consists of atoms as allowed in the normal selection and the logical operators ${\displaystyle \wedge }$ (and), ${\displaystyle \lor }$ (or) and ${\displaystyle \neg }$ (negation). This selection selects all those tuples in R for which φ holds.

To obtain a listing of all friends or business associates in an address book, the selection might be written as ${\displaystyle \sigma _{{\text{isFriend = true}}\,\lor \,{\text{isBusinessContact = true}}}({\text{addressBook}})}$. The result would be a relation containing every attribute of every unique record where isFriend is true or where isBusinessContact is true.

Rename (ρ)

A rename is a unary operation written as ${\displaystyle \rho _{a/b}(R)}$ where the result is identical to R except that the b attribute in all tuples is renamed to an a attribute. This is simply used to rename the attribute of a relation or the relation itself.

To rename the "isFriend" attribute to "isBusinessContact" in a relation, ${\displaystyle \rho _{\text{isBusinessContact / isFriend}}({\text{addressBook}})}$ might be used.

There is also the ${\displaystyle \rho _{x(A_{1},\ldots ,A_{n})}(R)}$ notation, where R is renamed to x and the attributes ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$ are renamed to ${\displaystyle \{A_{1},\ldots ,A_{n}\}}$.^[1]

Natural join (⋈)

Natural join (⋈) is a binary operator that is written as (R ⋈ S) where R and S are relations.^{[lower-alpha 1]} The result of the natural join is the set of all combinations of tuples in R and S that are equal on their common attribute names. For an example consider the tables Employee and Dept and their natural join:

Note that neither the employee named Mary nor the Production department appear in the result.

This can also be used to define composition of relations. For example, the composition of Employee and Dept is their join as shown above, projected on all but the common attribute DeptName. In category theory, the join is precisely the fiber product.

The natural join is arguably one of the most important operators since it is the relational counterpart of the logical AND operator. Note that if the same variable appears in each of two predicates that are connected by AND, then that variable stands for the same thing and both appearances must always be substituted by the same value (this is a consequence of the idempotence of the logical AND). In particular, natural join allows the combination of relations that are associated by a foreign key. For example, in the above example a foreign key probably holds from Employee.DeptName to Dept.DeptName and then the natural join of Employee and Dept combines all employees with their departments. This works because the foreign key holds between attributes with the same name. If this is not the case such as in the foreign key from Dept.Manager to Employee.Name then these columns must be renamed before taking the natural join. Such a join is sometimes also referred to as an equijoin (see θ-join).

More formally the semantics of the natural join are defined as follows:

${\displaystyle R\bowtie S=\left\{r\cup s\ \vert \ r\in R\ \land \ s\in S\ \land \ {\mathit {Fun}}(r\cup s)\right\}}$

(1)

where Fun(t) is a predicate that is true for a relation t (in the mathematical sense) iff t is a function (that is, t does not map any attribute to multiple values). It is usually required that R and S must have at least one common attribute, but if this constraint is omitted, and R and S have no common attributes, then the natural join becomes exactly the Cartesian product.

The natural join can be simulated with Codd's primitives as follows. Assume that c₁,...,c_m are the attribute names common to R and S, r₁,...,r_n are the attribute names unique to R and s₁,...,s_k are the attribute names unique to S. Furthermore, assume that the attribute names x₁,...,x_m are neither in R nor in S. In a first step the common attribute names in S can be renamed:

${\displaystyle T=\rho _{x_{1}/c_{1},\ldots ,x_{m}/c_{m}}(S)=\rho _{x_{1}/c_{1}}(\rho _{x_{2}/c_{2}}(\ldots \rho _{x_{m}/c_{m}}(S)\ldots ))}$

(2)

Then we take the Cartesian product and select the tuples that are to be joined:

${\displaystyle P=\sigma _{c_{1}=x_{1},\ldots ,c_{m}=x_{m}}(R\times T)=\sigma _{c_{1}=x_{1}}(\sigma _{c_{2}=x_{2}}(\ldots \sigma _{c_{m}=x_{m}}(R\times T)\ldots ))}$

(3)

Finally we take a projection to get rid of the renamed attributes:

${\displaystyle U=\Pi _{r_{1},\ldots ,r_{n},c_{1},\ldots ,c_{m},s_{1},\ldots ,s_{k}}(P)}$

(4)

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.

Car CarModel CarPrice CarA 20,000 CarB 30,000 CarC 50,000

Boat BoatModel BoatPrice Boat1 10,000 Boat2 40,000 Boat3 60,000

${\displaystyle {Car\bowtie Boat \atop \scriptstyle CarPrice\geq BoatPrice}}$ CarModel CarPrice BoatModel BoatPrice CarA 20,000 Boat1 10,000 CarB 30,000 Boat1 10,000 CarC 50,000 Boat1 10,000 CarC 50,000 Boat2 40,000

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 ${\displaystyle {R\ \bowtie \ S \atop a\ \theta \ b}}$ or ${\displaystyle {R\ \bowtie \ S \atop a\ \theta \ v}}$ where a and b are attribute names, θ is a binary relational operator in the set {<, ≤, =, ≠, >, ≥}, υ 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. It is important to note: 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 (⋉ and ⋊)

The left semijoin is a joining similar to the natural join and written as ${\displaystyle R\ltimes S}$ where ${\displaystyle R}$ and ${\displaystyle S}$ are relations.^{[lower-alpha 2]} 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:

${\displaystyle R\ltimes S=\{t:t\in R\land \exists s\in S(\operatorname {Fun} (t\cup s))\}}$

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

The semijoin can be simulated using the natural join as follows. If ${\displaystyle a_{1},\ldots ,a_{n}}$ are the attribute names of ${\displaystyle R}$, then

${\displaystyle R\ltimes S=\Pi _{a_{1},\ldots ,a_{n}}(R\bowtie S).}$

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.^[2]

Antijoin (▷)

The antijoin, written as R ▷ S where R and S are relations,^{[lower-alpha 3]} 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:

R ▷ S = R − R ⋉ S

(5)

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 ▷.

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. For an example see the tables Completed, DBProject and their division:

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:

R ÷ S = { t[a1,...,an] : t ∈ R ∧ ∀s ∈ S ( (t[a1,...,an] ∪ s) ∈ R) }

(6)

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.

Outer joins

Whereas the result of a join (or inner join) consists of tuples formed by combining matching tuples in the two operands, an outer join contains those tuples and additionally some tuples formed by extending an unmatched tuple in one of the operands by "fill" values for each of the attributes of the other operand. Outer joins are not considered part of the classical relational algebra discussed so far.^[4]

The operators defined in this section assume the existence of a null value, ω, which we do not define, to be used for the fill values; in practice this corresponds to the NULL in SQL. In order to make subsequent selection operations on the resulting table meaningful, a semantic meaning needs to be assigned to nulls; in Codd's approach the propositional logic used by the selection is extended to a three-valued logic, although we elide those details in this article.

Three outer join operators are defined: left outer join, right outer join, and full outer join. (The word "outer" is sometimes omitted.)

Operations for domain computations

There is nothing in relational algebra introduced so far that would allow computations on the data domains (other than evaluation of propositional expressions involving equality). For example, it is not possible using only the algebra introduced so far to write an expression that would multiply the numbers from two columns, e.g. a unit price with a quantity to obtain a total price. Practical query languages have such facilities, e.g. the SQL SELECT allows arithmetic operations to define new columns in the result SELECT unit_price * quantity AS total_price FROM t, and a similar facility is provided more explicitly by Tutorial D's EXTEND keyword.^[5] In database theory, this is called extended projection.^[6]: 213

Transitive closure

Although relational algebra seems powerful enough for most practical purposes, there are some simple and natural operators on relations that cannot be expressed by relational algebra. One of them is the transitive closure of a binary relation. Given a domain D, let binary relation R be a subset of D×D. The transitive closure R⁺ of R is the smallest subset of D×D that contains R and satisfies the following condition:

${\displaystyle \forall x\forall y\forall z\left((x,y)\in R^{+}\wedge (y,z)\in R^{+}\Rightarrow (x,z)\in R^{+}\right)}$

It can be proved using the fact that there is no relational algebra expression E(R) taking R as a variable argument that produces R⁺.^[7]

SQL however officially supports such fixpoint queries since 1999, and it had vendor-specific extensions in this direction well before that.

Selection

Rules about selection operators play the most important role in query optimization. Selection is an operator that very effectively decreases the number of rows in its operand, so if the selections in an expression tree are moved towards the leaves, the internal relations (yielded by subexpressions) will likely shrink.

Projection

Rename

Product and union

Cartesian product is distributive over union.

1. ${\displaystyle (A\times B)\cup (A\times C)=A\times (B\cup C)}$