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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 \(A\).


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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