Relation, and its types - Practice Questions & MCQ

Relation on a set
If a relation is from $A$ to $A$ itself, then this relation is called a relation on $\operatorname{set} A$.
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., $R=\varphi$

For example, Let $\mathrm{A}=\{2,4,6\}$ and $\mathrm{R}=\{(\mathrm{a}, \mathrm{b}): \mathrm{a}, \mathrm{b} \in \mathrm{A}$ and $\mathrm{a}+\mathrm{b}$ 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 $A \times A$
So, $R=A \times A$.
For example,
1. $\operatorname{Let} A=\{2,4\}$ and $R=\{(2,2),(2,4),(4,2),(4,4)\}$

Here, $R=A \times A$. 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 $\mathrm{A} \times \mathrm{A}$ 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 to $A$. It is denoted by $\mathrm{I}_{\mathrm{A}}$

$R=\{(a, b): a \in A, b \in A$ and $a=b\}$
It can also be written as $\mathrm{I}_{\mathrm{A}}=\{(\mathrm{a}, \mathrm{a}): \mathrm{a} \in \mathrm{A}\}$
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 $R_3$.

Note: In identity relation, all elements of type (a.a) should be there and there should not be any other element. But in reflexive relation, all elements of type $(\mathrm{a}, \mathrm{a})$ 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, $A=\{1,2,3\}$
$\mathrm{R}_1=\{(1,2),(2,1)\}$
$\mathrm{R}_2=\{(1,2),(2,1),(1,3),(3,1)\}$ 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
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 transitive relation on $A$ because $(2,3)$ is in $R_2$ and $(3,1)$ is in $R_2$ but $(2,1)$ is not in $R_2$.
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.