### Relational algebra & SQL

Query languages (computer languages used to make queries in databases) are abundant in style and usage today. In 2018, Microsoft released Azure Data Explorer to GA, introducing another query language that is specifically optimized for semi-structured, unstructured, and time-series data. Even though we have many querying language options, they are all grounded in a field of mathematics called relational algebra.

Relational Algebra is an algebra where operands are defined as relations, or variables that represent relations. This is why relational algebra provides the basis for a query language in relational databases.

Union $\cup$

Given relations $R$ and $S$, $R \cup S = \{T | T \in R \vee T \in S\}$. In SQL, this is written as

SELECT column1 [, column2 ]
FROM table1 [, table2 ]
[WHERE condition]

UNION

SELECT column1 [, column2 ]
FROM table1 [, table2 ]
[WHERE condition]


Intersection $\cap$

Given relations $R$ and $S$, $R \cap S = \{T | T \in R \wedge T \in S\}$. In SQL, this is written as

SELECT column1 [, column2 ]
FROM table1 [, table2 ]
[WHERE condition]

INTERSECT

SELECT column1 [, column2 ]
FROM table1 [, table2 ]
[WHERE condition]


Difference $\setminus$

Given relations $R$ and $S$, $R \setminus S = \{T | T \in R \wedge T \notin S\}$. In SQL, this is commonly written as

SELECT column1 [, column2 ]
FROM table1 [, table2 ]
[WHERE condition]

EXCEPT

SELECT column1 [, column2 ]
FROM table1 [, table2 ]
[WHERE condition]


However, a LEFT JOIN operator can also perform the same functionality.

Selection $\sigma$

Given relation $R$, $\sigma_{p}(R)$ is selecting typles that satisfy predicate $p = a \theta b$ or $p = a \theta v$ from a relation $R$. $a$ and $b$ are attribute names, $v$ is a value constant, and $\theta$ is a binary operation in $\{<. \le, =, \neq, \ge, >\}$. In SQL, this is simply the WHERE keyword. Given table R:

SELECT *
FROM R
[WHERE predicate condition]


Projection $\prod$

Given relation $R$, $\prod_{a_1, …, a_n}(R)$ is projecting a set obtained when the components of relation $R$ are restricted to the set $\{a_1, …, a_n\}$. In SQL, the SELECT operator performs this functionality. Given table R with attributes a, b, c

SELECT a, b, c
FROM R;


Fun Fact: the query language from Azure Data Explorer actually has a project keyword that behaves exactly like this. Given table R with attributes a, b, c:

R
| project a, b, c


Products $\times$

Given relations $R$ and $S$, $R \times S = \{(r,s) | r \in R and s \in S\}$. In SQL, this is known as a CROSS JOIN:

SELECT *
FROM table1
CROSS JOIN table2;


Natural Join $\Join$

Given relations $R$ and $S$, $R \Join S$ is the set of all combinations of tuples in $R$ and $S$ that are equal on their common attribute names. Formally:

$R \Join S = \{r \cup s | r \in R \wedge s \in S \wedge \text{Fun}(r \cup s)\}$

In SQL, this is known as the NATURAL JOIN:

SELECT *
FROM table1
NATURAL JOIN table2;


**Theta Join $\Join_{c}$**

Given relations $R$ and $S$, $R \Join_c S$ takes the product $R \times S$, then apply $\sigma_c$ to the result.

Renaming $\rho$

Given relation $R$, $\rho_{R(a_1, …, a_n}(R)$ makes the resulting relation be one with attributes $a_1, …, a_n$ and the same tuples as $R$. In SQL, the AS operator allows for remaining of attributes:

SELECT a AS apple, b AS banana
FROM R