Theory:

Many things around us in our daily lives are somehow related to one another through some kind of relation or connection.
Example:
1. Vijay and Ajith are friends, studying in the class. So here, the relation between Vijay and Ajith is "friends".
 
shutterstock_1700962303.jpg
 
2. Shreya went to the market and bought \(1 kg\) of wheat and \(2 kg\) of rice. Here, wheat and rice are the products, and it's purchased quantity are \(1 kg\) and \(2 kg\), respectively.
 
Therefore the relation are:
 
Wheat \(\longrightarrow\) \(1 kg\)
 
Rice \(\longrightarrow\) \(2 kg\)
 
wheat-2391348_1920.jpg
 
3. In chemistry, the atoms and its symbols, atomic number, and mass numbers are in relations.
 
For instance, the symbol of Nitrogen atom is "\(N\)", atomic number of Nitrogen is "\(7\)". Thus, the atom's symbol and its atomic number are related.
 
49827.jpg
Similarly, in mathematics, the term "relation' plays a vital role. We use the symbol "\(R\)" to represent the relation in mathematics.
 
Let's see some example to make it clear.
Example:
RelationshipRepresenting the relation using the symbol "\(R\)"Representation on ordered pair (Click! to know about ordered pairs)
Vinay is a student of Dr. VeenaVinay \(R\) Dr. Veena(Vinay, Dr. Veena)
Riya's sister is MeenaRiya \(R\) Meena(Riya, Meena)
Chennai is the capital of Tamil NaduChennai \(R\) Tamil Nadu(Chennai, Tamil Nadu)
\(-1\) is lesser than \(5\)\(-1\)  \(R\)  \(5\)\((-1, 5)\)
Square of \(9\) is \(81\)\(9\)  \(R\)  \(81\)\((9, 81)\)
 
From the above table, we can understand that how one thing is related to other. For example, how are Chennai and Tamil Nadu related?
 
We can say that, “Chennai is the capital of Tamil Nadu”. But apart from that there are several ways in which ‘Chennai' and ‘Tamil Nadu' are related. Which are:
  • Chennai is the capital of Tamil Nadu state.
  • Chennai is in the northern part of Tamil Nadu state.
  • Chennai is one of the largest cities of Tamil Nadu etc.,
So, when we want to specify a particular relation, providing only one ordered pair (Chennai, Tamil Nadu) it may not be practically helpful. But, if we ask the relation in the following set of ordered pairs, then specifying the relation is easy.
 
\(\text{\{(Chennai, Tamil Nadu), (Mumbai, Maharashtra), (Bengaluru, Karnataka), (Kolkata, West Bengal)\}}\)
Before we go any further into "relations," let's review the Cartesian product.
Cartesian product:
Let us consider the following two sets \(A\), \(B\).
 
A teacher conducts \(3\) different examinations for \(4\) students.
 
That is, the students \(A =\) \(\text{\{Kavya, Vimal, Raju, Nancy\}}\) and an examinations \(B =\) \(\text{\{English, Mathematics, Science\}}\)
 
Students \(A\)Examinations \(B\) 
Kavya \(k\)English \(e\)
Vimal \(v\)Mathematics \(m\)
Raju \(r\)Science \(s\)
Nancy \(n\) 
 
What are the possible ways of conducting an examination for the \(4\) students?
 
First, the four students write the English exam. That is, \(\{((k, e), (v, e), (r, e), (n, e)\}\). Similarly, if we apply the same method for other subjects such as Mathematics \((m)\) and Science \((s)\).
 
Mathematics \(=\) \(\{(k, m), (v, m), (r, m), (n, m)\}\)
 
Science \(=\) \(\{(k, s), (v, s), (r, s), (n, s)\}\)
 
All these subsets are a relation of \(A\) to \(B\).
 
Therefore, the Cartesian product:
 
\(A × B =\) \(\{(k, e), (v, e), (r, e), (n, e), (k, m), (v, m), (r, m), (n, m), (k, s), (v, s), (r, s), (n, s)\}\).
 
Let me ask you a question here.
 
If \(R_1 =\)  \(\{(k, s), (a, s)\}\), is this set have relations from \(A × B\)?
 
First, spot the subsets of \(R_1\) in the set of \(A × B\). If we could not find it then we can say that, \(\{(k, s), (a, s)\}\) \(∈ R_1\) but \(R_1 ∉ A × B\), because here the element \(a∉A\).
Reference: