JEE Main & Advanced Mathematics Sets Types of Relations

(1) 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 is reflexive \[\Leftrightarrow (a,\,\,b)\in R\] for all \[a\in A\].

Example : Let \[A=\{1,\,2,\,3\}\] and \[R=\{(1,\,\,1);\,\,(1,\,\,3)\}\]

Then R is not reflexive since \[3\in A\] but  \[(3,\,\,3)\notin R\]

A reflexive relation on A is not necessarily the identity relation on A.

The universal relation on a non-void set A is reflexive.

(2) Symmetric relation : A relation R on a set A is said to be a symmetric relation \[iff\,(a,b)\,\,\notin R\,\Rightarrow (b,a)\in R\] for all \[a,\,\,b\in A\]

i.e., \[aRb\Rightarrow bRa\] for all \[a,\,b\,\in A\].

it should be noted that R  is symmetric iff \[{{R}^{-1}}=R\]

The identity and the universal relations on a non-void set are symmetric relations.

A reflexive relation on a set A is not necessarily symmetric.

(3) Anti-symmetric relation : Let A be any set. A relation R on set A is said to be an anti-symmetric relation iff \[(a,\,\,b)\in R\]and \[(b,\,\,a)\in R\Rightarrow a=b\] for all \[a,\,\,b\in A\].

Thus, if \[a\ne b\] then a may be related to b or b may be related to a, but never both.

(4) Transitive relation : Let A be any set. A relation R on set A is said to be a transitive relation iff \[(a,\,\,b)\in R\] and \[(b,\,\,c)\in R\Rightarrow (a,\,\,c)\in R\] for all \[a,\,\,b,\,\,c\in A\] i.e.,  \[aRb\] and \[bRC\Rightarrow aRc\] for all \[a,\,\,b,\,\,c\,\in A\].

Transitivity fails only when there exists \[a,\,\,b,\,\,c\] such that a R b, b R c but \[a\,\not{R}\,c\].

Example : Consider the set \[A=\{1,\,\,2,\,\,3\}\] and the relations

\[{{R}_{1}}=\{(1,\,\,2),\,(1,\,3)\};\,\,{{R}_{2}}=\{(1,\,2)\};\,\,{{R}_{3}}=\{(1,\,\,1)\};\]

\[{{R}_{4}}=\{(1,\,\,2),\,\,(2,\,\,1),\,(1,\,\,1)\}\]

Then \[{{R}_{1}}\], \[{{R}_{2}}\], \[{{R}_{3}}\] are transitive while \[{{R}_{4}}\] is not transitive since in \[{{R}_{4}},\,(2,\,\,1)\in {{R}_{4}};\,(1,\,2)\in {{R}_{4}}\] but \[(2,\,2)\notin {{R}_{4}}\].

The identity and the universal relations on a non-void sets are transitive.

(5) 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. Every identity relation will be reflexive, symmetric and transitive.

Example : On the set \[=\{1,\,\,2,\,\,3\},\,\,R=\{(1,\,\,1),\,\,(2,\,\,2),\,\,(3,\,\,3)\}\] is the identity relation on A .

It is interesting to note that every identity relation is reflexive but every reflexive relation need not be an identity relation.

(6) 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. \[(a,\,\,b)\,\,\in R\Rightarrow (b,\,\,a)\in R,\] for all \[a,\,\,b\in A\]

(iii) It is transitive i.e. \[(a,\,\,b)\in R\] and \[(b,\,\,c)\in R\Rightarrow (a,\,\,c)\in R\] for all \[a,\,\,b,\,\,c\in A\].

Congruence modulo (m) : Let \[m\] be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo \[m\] if \[a-b\] is divisible by m  and we write \[a\equiv b\] (mod m).

Thus \[a\equiv b\] (mod m) \[\Leftrightarrow a-b\] is divisible by m. For example, \[18\equiv 3\] (mod 5) because \[18-3=15\] which is divisible by 5. Similarly, \[3\equiv 13\] (mod 2) because \[3-13=-10\] which is divisible by 2. But \[25\ne 2\] (mod 4) because 4 is not a divisor of \[25-3=22\].

The relation “Congruence modulo m” is an equivalence relation.