`(i) text( Empty Relation :)`

A relation R on a set A is said to be an empty relation or a void relation, if
`R= phi`.
For example,
Let `A= {1, 3}` and `R ={(a, b): a, b in A` and `a+ b` is odd}
Hence, R contains no element, therefore R is an empty relation on A.


`(ii) text(Universal Relation)`

A relation Ron a set A is said to be universal relation on A, if `R = A xx A.`
For example,
Let `A={1,2}`
and `R = [(1, 1), (1, 2), (2, 1), (2, 2)]`
Here, `R= A xx A`
Hence, R is the universal relation on A.

`(iii) text(Identity Relation)`

A relation Ron a set A is said to be the identity relation on A, if
`R = [(a, b): a in A, b in A` and `a= b]`
Thus, identity relation, `R = [(a, a): AA a in A]`
Identity relation on set A is also denoted by IA.
Symbolically, `I_A = [(a, a): a in A]`
For example,
Let `A= {1, 2, 3}`
Then, `I_A = {(1, 1), (2, 2), (3, 3)}`

`text(Note)`
In an identity relation on A every element of A should be related to itself only.

`(iv) text(Inverse Relation)`

If R is a relation from a set A. to a set B, then inverse relation of R to be denoted
by `R^(-1)`, is a relation from B to A
Symbolically, `R^(-1) = {(b, a): (a, b) in R}`
Thus, `(a, b) in R ⇔ (b,a) in R^(-1) ,AA a in A , b in B`

For example,
If `R = {(1, 2), (3, 4), (5, 6)},` then
`R^(-1) ={(2,1),(4,3),(6,5)}`
`therefore (R^(-1))^(-1) = {(1, 2), (3, 4), ( 5, 6)} = R`
Here, dom `(R) = { 1, 3, 5},` range `(R) = { 2, 4, 6}`
and dom `(R^(-1)) = {2, 4, 6},` range `(R^(-1)) = {1, 3, 5}`
Clearly, dom `(R^(-1)) =` range (R)
and range `(R^(-1)) =` dom `(R)`.

` text(Identity Relation) :-`

A relation defined on a set `A` is said to be an Identity relation if every element of `A` is related to itself and
only to itself.

`Eg. 1` `A` relation defmed on the set of natural number's as

with rule `aRb <=> a = b` is an identity relation

`R = {(1, 1), (2, 2), (3, 3), ......... }`

Eg: 2 The relation l `A = { (1 ,1), (2, 2), (3, 3 ), ......... }` is the identity relation on set `A = {1 , 2, 3}` but

`{(1,1), (2, 2), (1, 3)}` are not identity relation

` text(Reflexive):-`

A relation defined on a set `A` is said to be an identity relation if each & every element ofA is related to
itself.

if `(a, b) in R` then `(a, a) in R`. However if there is a single ordered pair of `(a, b) in R` such
`(a, a) !in R` then `R` is not reflexive.

Eg. `1 `: Let `A = {1, 2, 3}` be a set then `R = {(1, 1),(2,2), (3, 3), (1,3),(2, 1)}` is a reflexive relation on `A.`

`R_1 = { (1,1), (3, 3), (2, 1), (6, 2)}` is not a reflexive relation on `A,` because `2 in A` but `(2, 2) in R.`

E.g. `2 : A` relation defined on (set of natural numbers)

`aRb => 'a'` divides `'b' a, b in N`

`R` would always contain `(a, a)` because every natural number divides itself and hence it is a reflexive
relation.

`text( Note):-`
Every identity relation is a reflexive relation but every reflexive relation need not be an identity.


`text(Symmetric Relation):-`

A relation defined on a set is said to be symmetric

if `aRb => bRa.` If `(a, b) in R` then `(b, a)` must be necessarily there in the same relation.

Eg:-

(i) `a R b => a` is parallel to `b`
It is a symmetric relation because if `a` is parallel to `b` then the line `b` is parallel to `a.`

`(ii)` `L_1 R L_2 ........ L_1` is perpendicular to `L_2` is a symmetric relation.

`(iii)` `a R b => a` is brother of `b` is not necessarily brother `a.`

`(iv)` `a R b => a` is a cousin of `b`. This is a symmetric relation.

`text(Note) :`
lf `R` is symmetric

`(i)` `R = R^(-1)`

`(ii)` Rangle of `R =` Domain of `R`


`text(Anti-symmetric Relation : )`

A relation Ron a set A is said to be anti-symmetric,
if `a R b, b R a => a= b, V a, b in A`
i.e., `(a, b) in R` and `(b, a) in R => a = b, AA a, b in A`
For example,
Let R be the relation in N (natural number) defined by, "x is divisor of y,"
then R is anti-symmetric because x divides y andy divides `x => x = y`



` text( Transitive relation :)`

A relation on set `A` is said to be transitive if `aRb` and `bRc` implies `aRc` then it is transitive.

`(a, b)` in `R & (b, c)` in `R => (a, c) in R` and `(a, b, c)` need not be distinct.

`Eg. 1 : aRb (a - b)` is even

`(6, 4), (4, 20) => (6, 20) in R`

`Eg. 2`: On the set of natural numbers, the relation `R` defined by `x R y => x < y` because for any

`x , y, z in N` `x < y, y < z => x < z.`

`text(Equivalence Relation)`

A relation R on a set A is said to be an equivalence relation on A. when R is
(i) reflexive (ii) symmetric and (iii) transitive. The equivalence relation denoted by ~


`text(Ordered Relation)`
A relation R is called ordered, if R is transitive but not an equivalence
relation.
Symbolically,` a R b, b R c => a R c, AA a, b, c in A`

`text(Partial Order Relatiion)`

A relation R is called partial order relation, if R is reflexive, transitive and
anti-symmetric at the same time.
For example,
Let `R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}`
`therefore R^(-1) = {(1, 1), (2, 2), (3, 3), (2, 1), (3, 2), (3, 1)}`
`R nn R^(-1) = {(1, 1), (2, 2), (3, 3)} =` Identity
`therefore` R is anti-symmetric.
It is clear that R is reflexive.
Since, `(1, 1) in R, (2, 2) in R, (3, 3) in R` and R is transitive.
Since, `(1, 2) in R` and `(2, 3) in R => (1, 3) in R`
Hence, R is partial order relation.