16.2.1. Definition of Relation

So, what is a relation? So, let me begin with this example, you consider this table and people who are familiar with databases, they very well know what is a table. Table basically consist of several columns and each column have some entries. So, I call this table T which has two columns, column number 1 and column number 2. In column number 1, you have some countries and in column number 2, you have some cities listed.

Now if your general knowledge is good then it turns out that the elements you can spot here that elements of the first column and the second column they are related by some relationship. And the relationship here is that, in the first column I have listed some of the countries and in the second column I have listed the capital of the corresponding countries.

A relation is a way to connect two sets of items. In this case, we have a table with two columns: one for countries and another for their corresponding capital cities. Each row in the table demonstrates a relationship between an entry in the first column and the second column, particularly showing which city is the capital of which country. Thus, we can think of a relation as a meaningful connection that helps us understand how elements from two different sets correlate with each other.

Think about a school roster where one column has the names of students and another has their respective grades. Each student's name is related to their grade, just like the countries are related to their capitals. This helps us understand performance at a glance, much like the country-capital relationship helps us identify governance structures.

Now how do we mathematically interpret this table? Is there any mathematical interpretation or mathematical abstraction by which you can define this table? Well the way I can mathematically interpret this table T is as follows. I can imagine that I have a set A which is defined to be the set of all countries in the world. Well I do not know right now how many exact numbers of countries in the world definitely it is more than 200. So, A has more than 200 elements. Whereas B is another set, which is defined to be the set of all cities in the world.

In mathematics, we can define the relationship shown in the table more formally using sets. Let’s imagine two sets: set A includes all the countries and set B includes all the cities. By creating a relationship between these sets, we can form pairs that link countries to their respective capitals. This allows us to take a much larger structure (which could contain many different relationships) and focus on a specific subset where each country is paired with its capital city.

Imagine you have two boxes filled with different toys (set A being action figures and set B being playsets). By relating some action figures to certain playsets, you're forming specific connections (relations) just like we relate countries to their capitals. Instead of all toys, we can focus only on the specific ones that match perfectly.

Now if I take the Cartesian product of A and B, what will I obtain? The Cartesian product of A and B will be a set of the form (a, b), where a will be some country. Namely, it will be belonging to A and b will be some city. As of now when I take A x B, there is no relationship between the elements a,b, I am just picking some country and some city, country, city. I have listed down all possible pairs of the form country, city and this will be an enormously large set.

The Cartesian product of two sets, A (countries) and B (cities), generates all possible paired combinations of the elements from both sets. This means from our earlier example, it will result in every possible pairing of country with city, regardless of whether the city is the capital of the country or not. From this large set, we identify and extract a smaller, specific subset where each country is paired exclusively with its correct capital, which is our relation.

If you think of putting together a puzzle of a world map (countries), and a separate box of city names (cities), every possible combination of the countries and cities is like generating the vast Cartesian product. However, only the cities correctly labeled as capitals would fit perfectly with the countries, just like finding the right pieces of a puzzle.

So, that is a loose definition of a relation. A relation here is basically a subset of A x B, if I am considering two sets A and B and of course whatever I have discussed here can be extended where I have multiple sets. What do I mean by that? In this example I had only two columns...this will be considered as a subset of the Cartesian product of all the big sets A, B, C, D from which the elements in your column C1, C2, C3, C4 are occurring here.

In a more formal mathematical context, a relation is defined as a subset of the Cartesian product of two sets A and B. This means that not all combinations from A and B will qualify as a relation, but only those combinations that align with a specific, meaningful context (like country-capital pairs). Furthermore, this concept can be extended to multiple sets, where relations can account for additional columns (or attributes), creating more complex relationships across multiple datasets.

Think of a dating app where profiles (set A with names) are related to potential matches (set B with their interests). The actual matches are really a subset of all possible combinations of profiles and interests, much like how a specific relation emerges from all combinations formed in our country-capital example.

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