Relation (Mathematics) - Notes, Concept and All Important Formula

 RELATIONS

1. INTRODUCTION :

Let $A$ and $B$ be two sets. Then a relation $R$ from $A$ to $B$ is a subset of $A \times B$. thus, $R$ is a relation from $A$ to $B \Leftrightarrow R \subseteq A \times B$.

Total Number of Relations : Let $A$ and $B$ be two non-empty finite sets consisting of $m$ and $n$ elements respectively. Then $A \times B$ consists of mn ordered pairs. So total number of subsets of $A \times B$ is $2^{m n}$.

Domain and Range of a relation : Let $R$ be a relation from a set $A$ to a set $B$. Then the set of all first components or coordinates of the ordered pairs belonging to $R$ is called to domain of $R$, while the set of all second components or coordinates of the ordered pairs in $R$ is called the range of $R$.

Thus, $\quad$ Domain $( R )=\{ a :( a , b ) \in R \}$

and, Range $( R )=\{ b :( a , b ) \in R \}$

It is evident from the definition that the domain of a relation from $A$ to $B$ is a subset of $A$ and its range is a subset of $B$.

Inverse Relation : Let A, B be two sets and let $R$ be a relation from a set $A$ to a set $B$. Then the inverse of $R$, denoted by $R ^{-1}$, is a relation from $B$ to $A$ and is defined by

$R ^{-1}=\{( b , a ):( a , b ) \in R \}$

Clearly, $\quad( a , b ) \in R \Leftrightarrow( b , a ) \in R ^{-1}$

Also, $\quad$ Domain $( R )=$ Range $\left( R ^{-1}\right)$ and Range $( R )=\operatorname{Domain}\left( R ^{-1}\right)$

Note : Relation on a set : If $R$ is a relation from set $A$ to $A$ itself then $R$ is called Relation on set $A$.

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All Chapter Notes, Concept and Important Formula

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2. TYPES OF RELATIONS :

In this section we intend to define various types of relations on a given set $A$.

Void Relation : Let $A$ be a set. Then $\phi \subseteq A \times A$ and so it is a relation on A. This relation is called the void or empty relation on 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 the universal relation on 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 $A$. In other words, a relation $I _{ A }$ on $A$ is called the identity relation if every element of A is related to itself only.

Reflexive Relation : A relation $R$ on a set $A$ is said to be reflexive if every element of $A$ is related to itself.

Thus, $R$ on a set $A$ is not reflexive if there exists an element $A \in A$ such that $( a , a ) \notin R$.

Every Identity relation is reflexive but every reflexive relation is not identity.

Symmetric Relation : A relation $R$ on a set $A$ is said to be a symmetric relation iff

$(a, b) \in R \Leftrightarrow(b, a) \in R$

i.e. $\quad a R b \Leftrightarrow bRa$

Transitive Relation : Let $A$ be any set. A relation $R$ on $A$ is said to be a transitive relation iff

$(a, b) \in R$ and $(b, c) \in R \Rightarrow(a, c) \in R$

i.e. $\quad$ a $R\,\, b$ and $b R c \Rightarrow a R c$

Antisymmetric Relation : Let $A$ be any set. A relation $R$ on set $A$ is said to be an antisymmetric relation iff

$(a, b) \in R$ and $(b, a) \in R \Rightarrow a=b$

Equivalence Relation : A relation $R$ on a set $A$ is said to be an equivalence relation on $A$ iff

(i) it is reflexive i.e. $(a, a) \in R$ for all $a \in A$

(ii) it is symmetric i.e. $( a , b ) \in R \Rightarrow( b , a ) \in R$

(iii) it is transitive i.e. $(a, b) \in R$ and $(b, c) \in R \Rightarrow(a, c) \in R$

It is not necessary that every relation which is symmetric and transitive is also reflexive.

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