Domain and Range of a Relation

A relation in mathematics is a connection or association between elements of two sets. This is crucial for checking the common terms or anything between two or more functions. The type of relations signifies its types and the type of relations used in different aspects of different functions. The main applications of these types of relations are in database systems, graph theory, and social networks.

In this article, we will cover the concept of relations and its types. This concept falls under the broader category of sets relation and function, a crucial Chapter in class 11 Mathematics. It is not only essential for board exams but also for competitive exams like the Joint Entrance Examination (JEE Main), and other entrance exams such as SRMJEE, BITSAT, WBJEE, BCECE, and more. The types of relation is very crucial for mathematics.

Introduction to relations

Relation is defined as the relation between two different sets of information. Suppose we are given two sets containing two different values then a relation defined such that it connects the value of the first set with the value of the second set is called the relation.

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the cartesian product $A$ $\times B$.

The subset is derived by describing the relationship between the first element and the second element of the ordered pairs in $A \times B$.

The second element is called the image of the first element.
If the element $(a, b)$ belongs to $R$, (here $a$ belongs to $A$ and $b$ belongs to $B$ ), then the relation is represented as a $R b$.

Example: The relation $R=\{(1,1),(2,4),(3,9)\}$ and it is represented using $\}$.
Relation on a set
If a relation is from $A$ to $A$ itself, then this relation is called a relation on set $A$.
Types of relation
Empty Relation
A relation $R$ on a set $A$ is called an empty relation, if no element of $A$ is related to any element of A, i.e., $\mathrm{R}=\varphi$

For example, Let $A=\{2,4,6\}$ and $R=\{(a, b): a, b \in A$ and $a+b$ is odd $\}$
Here, $R$ contains no element, therefore $R$ is an empty relation on $A$

Universal Relation
A relation $R$ on a set $A$ is called a universal relation, if each element of $A$ is related to every element of $A$, i.e., $R$ has all the ordered pair contained in $\mathrm{A} \times \mathrm{A}$

So, $R=A \times A$.
For example,
1. Let $A=\{2,4\}$ and $R=\{(2,2),(2,4),(4,2),(4,4)\}$

Here, $\mathrm{R}=\mathrm{A} \times \mathrm{A}$. Hence, R is a Universal relation
2. Let $A=\{1,2,3\}$, and $R=\{(a, b):|a-b|>-2, a, b \in A\}$

Clearly the mod value of the difference of any pair $(a, b)$ will be greater than -2
So, each possible ordered pair in $A \times A$ will lie in $R$, therefore $R$ is a universal relation
Identity relation
If every element of $A$ is related to itself only, then it is known as an identity relation on $A$. It is denoted by $\mathrm{I}_{\mathrm{A}}$

$$
R=\{(a, b): a \in A, b \in A \text { and } a=b\}
$$
It can also be written as $I_A=\{(a, a): a \in A\}$

For example A = {2 ,4, 6}

Then, $\mathrm{I}_{\mathrm{A}}=\{(2,2),(4,4),(6,6)\}$

Reflexive Relation

A relation $R$ on a set $A$ is called Reflexive, if $(a, a) \in R$, for every $a \in A$,
For example: let $A=\{1,2,3\}$
- $\mathrm{R}_1=\{(1,1),(2,2),(3,3)\}$
- $\mathrm{R}_2=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3)\}$
- $\mathrm{R}_3=\{(1,1),(2,2),(2,3),(3,2)\}$

Here $R_1$ and $R_2$ are reflexive relations on $A, R_3$ is not a reflexive relation on $A$ as $(3,3)$ is not present in $\mathrm{R}_3$.

Note: In identity relation, all elements of type (a.a) should bel there and there should not be any other element. But in reflexive relation, all elements of type $(a, a)$ should be there, and apart from these, other elements can also be there

So, $R_1$ is an identity relation and is also reflexive, but $R_2$ is only reflexive and not an identity relation

Symmetric Relation
A relation $R$ on a set $A$ is said to be a symmetric relation, if a $R b \Rightarrow b R a, \forall a, b \in A$
For example, $A=\{1,2,3\}$
- $\mathrm{R}_1=\{(1,2),(2,1)\}$
- $R_2=\{(1,2),(2,1),(1,3),(3,1)\}$ and
- $\mathrm{R}_3=\{(2,3),(1,3),(3,1)\}$

Here $R_1$ and $R_2$ are symmetric relations on $A$ but $R_3$ is not a symmetric relation on $A$ because $(2,3)$ is in $R_3$ and $(3,2)$ is not in $R_3$.

Transitive Relation

|Transitive Relation
A relation $R$ on a set $A$ is said to be a transitive relation, if a $R b$ and $b R c \Rightarrow a R c, \forall a, b, c \in A$
For example, Let $A=\{1,2,3\}$
- $\mathrm{R}_1=\{(1,2),(2,3),(1,3),(3,2)\}$
- $\mathrm{R}_2=\{(2,3),(3,1)\}$
- $\mathrm{R}_3=\{(1,3),(3,2),(1,2)\}$

Here $R_1$ is not a transitive relation on $A$ because $(2,3)$ is in $R_1$ and $(3,2)$ is in $R_1$ but $(2,2)$ is not in $R_1$. Also, $(3,2)$ in $R_1$ and $(2,3)$ is in $R_1$, but $(3,3)$ is not in $R_1$

Again $R_2$ is not a transitive relation on $A$ because $(2,3)$ is in $R_2$ and $(3,1)$ is in $R_2$ but $(2,1)$ is not in $\mathrm{R}_2$.

Finally, $R_3$ is a transitive relation.
A relation $R$ on a set $A$ is said to be an equivalence relation if $R$ is reflexive, symmetric, and transitive.

Inverse Relations
Inverse relation occurs when a set has inverse pairs of another set. i.e. if $R \in A \times B$ then the inverse relation is $R^{-1}=\{(b, a)$ such that $(a, b) \in R\}$

Consider if set $A=\{(1,2),(3,4)\}$, then inverse relation will be $R-1=\{(2,1),(4,3)\}$.

^Summary

We concluded that the understanding of relations is fundamental in discrete mathematics, computer science, and related fields, as it lays the groundwork for more complex structures and analyses. This helps in a lot of applications that are used in our day-to-day life.

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Solved Examples Based On the Relation and its types:

Example 1: If set $A=\{1,2,3,4,5\}$, then which of the following is a void relation from $A \rightarrow A$ ?
1) $\{(a, b): a \times b=6\}$
2) $\{(a, b): a \times b=15\}$
3) $\{(a, b): a \times b=40\}$
4) $\{(a, b): a \times b=16\}$

Solution:
$a \times b=40$ is not possible, since no two numbers ( $a, b$ ) in A can give the product 40 .
So, it is a void relation.
Hence, the answer is the option 3.
Example 2: If $A \times A=\{1,1\}$, then it is a subset of
1) $A \cup A$
2) $A \cap A$
3) $A \times A$
4) $A-A$

Solution:

As we learned

Universal Relation -

Self-product of a set

wherein

$A X A \subseteq A X A$
Clearly. $A=\{1\}$
Thus $A \times A \subseteq A \times A$
Hence, the answer is the option 3.
Example 3: Which of the following relations is an identity relation in set $A=\{1,2,3\}$ ?
1) $\{(1,1),(1,2),(1,3),(2,2),(3,3)\}$
2) $\{(1,1),(2,2),(3,3),(2,3)\}$
3) $\{(1,1),(2,2),(3,3)\}$
4) All of the above.

Solution:
Only (3) is an identity relation, Since all of (1,1), (2,2), and (3,3) are present. Also, there are no extra elements present in it.
Hence, the answer is the option 3.
Example 4: Which of the following relations is both reflexive and an identity relation on $\operatorname{set} A=\{a, b\}$ ?
1) $\{(a, b),(a, a),(b, b)\}$
2) $\{(a, a),(b, b)\}$
3) $\{(a, b),(b, a),(a, a),(b, b)\}$
4) All of the above.

Solution:

Though all the above are reflexive relations, but only {(a, a),(b,b)} is an identity relation on A.

Hence, the answer is the option 2.

Example 5: If set A = {a,b,c,d}, then which of the following is a relation on set A?

1) $\{(a, e),(b, d)\}$
2) $\{(a, a),(d, c)\}$
3) $\{(b, b),(c, p)\}$
4) $\{(b, c),(c, f)\}$

Solution:

In any relation on set A, both elements of the ordered pair should belong to A.

Only the relation in option (2) satisfies this condition, so it is a relation on set A.

Hence, the answer is the option 2.

Frequently Asked Questions(FAQ)-

1. What is relation?

Ans: A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the cartesian product $\mathrm{A} \times \mathrm{B}$.
2. What is a number of relations?

Ans: The number of possible relations between two sets $A$ and $B$ depends on the sizes of these sets
3. What is the difference between reflexive and transitive relations?

Ans: A relation $R$ on a set $A$ is called Reflexive, if $(a, a) \in R$, for every $a \in A$ whereas a relation $R$ on a set $A$ is said to be $a$ transitive relation, if a $R b$ and $b R c \Rightarrow a R c, \forall a, b, c \in A$
4. What is the difference between empty and universal relation?

Ans: A relation $R$ on a set $A$ is called an empty relation, if no element of $A$ is related to any element of $A$, i.e., $R=\varphi$ whereas a relation $R$ on a set $A$ is called a universal relation, if each element of $A$ is related to every element of $A$, i.e., $R$ has all the ordered pairs contained in $A x$ A.
5. How do you define reflexive relations?

Ans: A relation $R$ on a set $A$ is called Reflexive, if $(a, a) \in R$, for every $a \in A$.