Relations in Maths: Definition, Types, Examples & Diagram

A relation defines the connection between two sets' elements in maths. With it, you can map the elements of one set (domain) to elements of another set (range) such that the resulting ordered pairs are of the form (input, output).

Here is how you can instantly apply the concept of relations!

What they do - Relations show how two different objects coexist. They define a type of relationship between two objects. 

How do we represent relations - Their usual mathematical representation is of an ordered pair - two-numbered pairs in brackets. For instance, you have (input, output) or (x, y). We call them the set's elements.

Also, you will find that there are different types of relations. They establish a connection between two quantities. They are functions. When you read more about functions, you will find that they are subsets of relations.   

Importance of Relations

1. Relations are integral to learning maths, data analytics, and computer science at higher levels. They are also essential for your JEE Mains. 
2. Understanding relations helps students like you decipher patterns, connections, and behaviour making you better at problem-solving.

The NCERT definition of relation in Math is in Chapter 2 of NCERT Maths textbook. 

“A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B. The second element is called the image of the first element.”

The explanation goes like this. The basic definition of relations in maths is that they are a subset of the Cartesian product of two sets. To recap your knowledge, a Cartesian product is a mathematical operation. It returns a set of all ordered pairs from two sets.

Let's say there are two sets as 𝑋 and 𝑌. 

Let 𝑥∈𝑋(𝑥 is an element of the set 𝑋) and 𝑦∈𝑌. 

We show X and Y's Cartesian product is shown as X×Y. 

We define it by collecting all possible ordered pairs (x, y).

Simply put, a relation will show that every input will create one or more outputs.

Consider we have two sets 𝑋={4,36,49,50} and 𝑌={1,−2,−6,−7,7,6,2}. So, a relation R includes the pair (x, y) if x is the square of y. Now that can be highlighted when using ordered pairs as 𝑅={(4,−2), (4,2), (36,−6), (36,6), (49,−7), (49,7)}.

Relations can be represented using different techniques. There are five main representations of relations.

Set Builder Form

It is a mathematical notation where the rule that associates the two sets X and Y is clearly specified. If there are two sets $X=\{\mathrm{5,6},7\}$ and $Y=\left\{\mathrm{25,36,49}\right\}$ . The rule is that the elements
of X are the positive square root of the elements of Y . In set-builder form this relation can be written as $R=\left\{\right(a,b)$ : $a$  is the positive square root of $b,a\in X,b\in Y\}$ .

Roster Form

In roster form, all the possible ordered pairs of the two sets that follow the given relation are written. Using the same example, the relation is represented as $R=\left\{\right(\mathrm{5,25}),(\mathrm{6,36}),(7$ , 49)}.

Such a diagram is used to visually represent the relation between the elements of the two given sets.

Tabular Form

When the input and the output of a relation are expressed in the form of a table:
X           Y
$5\mathrm{}25$

$6\mathrm{}36$
7            49

Different kinds of relations are needed to classify the connections between two sets.

Empty Relation: An empty relation is one where no element of a set is mapped to another or itself. Example: $R=\varnothing$ .

Universal Relation: If all elements of one set are mapped to all elements of another set or itself.

Identity Relation: Each element of a set is related to itself. Example: $\mathrm{I}=\left\{\right(\mathrm{x},\mathrm{x})$ : for all $\mathrm{x}\in \mathrm{X}\}$ .

Inverse Relation: Inverse of a relation $R$ is $R^{-1}=\left\{\right(y,x):(x,y)\in R\}$ .

Reflexive Relation: All elements map to themselves: . $(\mathrm{x},\mathrm{x})\in \mathrm{R}$
Symmetric Relation: If $(x,y)\in R$  then $(y,x)\in R$ .
Transitive Relation: If $(y,x)\in R$ and $(y,z)\in R$  then $(x,z)\in R$ .

Equivalence Relation: A relation that is symmetric, reflexive, and transitive.

One to One Relation: Each element of one set maps to a distinct element of another.

One to Many Relation: A single element maps to multiple elements in another set.

Many to One Relation: Multiple elements map to a single element in another set.

Many to Many Relation: Multiple elements map to multiple elements across sets.

All these type of Relations are also included in our NCERT Class 12 Relations and Functions Solutions, and good number of practice questions are given for better preparations.

Relations can also be represented graphically using the cartesian coordinate system. The ordered pair represents the position of points in a coordinate plane. For instance, a relation $\mathrm{y}=\mathrm{x}-2$ can be plotted by substituting x with values such as $-\mathrm{1,0}$ , and 2 , resulting in y values of -3 , -2, and 0 respectively. These ordered pairs are plotted to form a graph, which in this case would be a straight line.