Im putting a technical presentation for an interview (topic I chose). I am researching real-time data streaming. I am familiar with what an ordered pair is as it pertains to a graph i.e (x, y) defines a point on the x,y axis.
May you explain to me how this relates to columns in a database? Does this have to do with the index?
I'm trying to visualize this but all I can think about is result sets in the DBMS and want to learn how this applies.
Thanks
The idea you're looking for is the relation. Relational databases are an application of the mathematical theory of relations.
If you have two sets, $A$ and $B$, you can form the set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. We call this set of ordered pairs the Cartesian product of $A$ and $B$, denoted $A \times B$. In other words
$$A \times B = \{(a,b) : a \in A, b \in B \}$$
For example if $A = B = \mathbb{R}$, then $A \times B = \mathbb{R} \times \mathbb{R} = \mathbb{R}^2$.
This is a set-theoretical way to think of the Cartesian plane as the product of the $x$- and $y$-axes.
Another example would be if $A = \{$apple, banana} and $B = \{$yellow, red} then
$A \times B =$ {(apple, yellow), (apple, red), (banana, yellow), (banana, red)}.
Now, what is a relation on $A$ and $B$? It's a subset of their Cartesian product.
For example if we wish to specify the relation "equals" on $\mathbb{R} \times \mathbb{R}$ we would specify the subset of $\mathbb{R}^2$ consisting of exactly those pairs $(x,y)$ for which $x = y$.
So you see that we can express a relation as a subset of a Cartesian product.
In our fruit/color database, the relation of interest is the set of ordered pairs representing each fruit and its actual color. This is represented by the subset {(apple, red), (banana, yellow)}.
The key point is that A relation between $A$ and $B$ is a subset of $A \times B$.
In the general case, suppose you have $n$ sets $S_1, S_2, \dots, S_n$. Then a relation on these $n$ sets is just a subset of the Cartesian product of all of them, where instead of ordered pairs, the Cartesian product is the set of all ordered n-tuples formed from the sets.
You can now see that if you have a database table representing the employees at a company, the data in the database is a subset of the Cartesian product of sets representing the employee name, employee number, job title, salary, years on the job, etc.
Once we choose a subset of that Cartesian product, we have a database table populated with the employees and their respective information.
The rest of relational database theory is worked out in terms of relational algebra on sets.
To sum this up: A relation is a subset of the Cartesian product of a collection of $n$ sets; each set representing a column in a database table. The contents of the database are a particular set of $n$-tuples, each $n$-tuple representing a row of the database table.