Functions
3.0 Types of Relation
3.0 Types of Relation
Void relation:
Let $A$ be a set. Then, $\emptyset \subseteq A \times A$ and so it is a relation on $A$. This relation is called void or empty relation on set $A$.
Universal relation:
Let $A$ be a set. Then, $A \times A \subseteq A \times A$ and so it is a relation on $A$. This relation is called universal relation on set $A$.
A relation $R$ on a set is called universal relation, if each element of $A$ is related to every element of $A$.
Identity relation:
Let $A$ be a set. Then, the relation ${I_A} = \{ (a,a):a \in A\} $ on $A$ is called the identity relation on $A$. A relation ${I_A}$ on set $A$ is called the identity relation, if every element of $A$ is related to itself only.
For example, if $A = \{ 1,2,3\} $, then the relation ${I_A} = \{ (1,1),(2,2),(3,3)\} $ is the identity function on set $A$. But the relation ${R_1} = \{ (1,1),(3,3)\} $ and ${R_2} = \{ (1,1),(2,2),(3,3),(2,3)\} $ are not identity functions as $(2,2) \notin {R_1}$ and in ${R_2}$ the element $1$ is related to two other elements, $1$ and $3$.
Reflexive relation:
A relation $R$ on a set $A$ is said to be reflexive if every element of $A$ is related to itslef. $$R\;is\;reflexive\; \Leftrightarrow (a,a) \in R\;for\;all\;a \in A$$
For example, Let set $A$ be defined as $A = \{ a,b,c\} $ . Then, ${R_1} = \{ (a,a),(b,b),(c,c),(a,c)\} $ is reflexive while ${R_2} = \{ (a,a),(a,b),(b,a),(c,c)\} $ is not reflexive as $b \in A$ but $(b,b) \in {R_2}$
Symmetric relation:
A relation $R$ on a set $A$ is said to be symmetric relation iff, $$\begin{equation} \begin{aligned} (a,b) \in R \Rightarrow (b,a) \in R\;\;for\;\;all\;\;a,b \in A \\ aRb \Rightarrow bRa\;\;for\;\;all\;\;a,b \in A \\\end{aligned} \end{equation} $$
For example, Let set $A$ be defined as $A = \{ a,b,c\} $ . Then, ${R_1} = \{ (a,a),(b,b),(c,c),(a,c)\} $ is not symmetric as $(a,c) \in {R_1}$ but $(c,a) \notin {R_1}$ while ${R_2} = \{ (a,a),(a,b),(b,a),(c,c)\} $ is symmetric.
Transitive relation:
Let $A$ be any set. A relation from $R$ on $A$ is said to be a transitive relation iff $$\begin{equation} \begin{aligned} (a,b) \in R\;\;and\;\;(b,c) \in R \Rightarrow (a,c) \in R\;for\;all\;a,b,c \in A \\ aRb\;and\;bRc \Rightarrow aRc \\\end{aligned} \end{equation} $$
Antisymmetric relation:
Let $A$ be any set. A relation from $R$ on $A$ is said to be antisymmetric relation iff $$(a,b) \in R\;\;and\;\;(b,a) \in R \Rightarrow a = b\;\;\;for\;all\;a,b \in A$$
Note: It follows from this definition that if $(a,b) \in R$ but $(b,a) \notin R$ , then also $R$ is an antisymmetric relation.
Equivalence relation:
A relation $R$ on a set $A$ is said to be an equivalence relation on $A$ iff it is
$$\begin{equation} \begin{aligned} (i)\;reflexive\; \\ (ii)\;symmetric \\ (iii)\;transitive \\\end{aligned} \end{equation} $$
