What is Relation ? What are the types of Relaions ?

Relation :  A relation on A is the subset of A*A .

Let A = {0,1}

A*A={(0,0),(0,1),(1,0),(1,1)}

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The total number of subset of Any Set = 2^n

So Subset of A*A = 2^4=16

List the all 16 relations on {0,1}

Types of Relation :

Reflexive

Irreflexive

Symmetric

Antisymmetric

Asymmetric

Transitive

Reflexive : A relation on Set A is called Reflexive

For all a belongs to A

(a,a) belongs to R

Key Point : For Reflexive Relation a Relation must contain non null main diagonal elements

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E.g  just like given table main diagonal elements are (0,0) (1,1)

1. {} Not reflexive Because the diagonal elements are null
2. A*A yes because it contain main diagonal elements
3. {( 0,0),(1,1) } yes for all a belongs to A there is (a,a) belongs to R
4. { (0,0) ,(1,0),(1,1) } yes because no diagonal element is null
5. { (0,0) ,(1,0)} no because one diagonal element (1,1) is null

Irreflexive :  A relation R on A is said to be Irreflexive

If for all a belongs to A (a,a) does not belongs to R it requires NULL Diagonal Elements

1. {} yes because there is no pair (a,a) belongs to R
2. A*A no because there is pair (a,a) belongs to R
3. { (1,0),(0,1)} yes as the diagonal elements are null
4. {(1,1),(0,1} no because all main diagonal elements are not null

Symmetric : A relation on A is said to be Symmetric

For all a and b belongs to A

If There is (a,b) belongs to R

Then (b,a) must belongs to R

1. {} yes because there is no pair of (a,b) so we have to not check for (b,a)
2. A*A yes because for any pair of (a,b) there is (b,a)
3. {(1,1),(2,2),(3,3) } yes
4. { (1,1) ,(2,3),(3,2)} yes because for each pair of (a,b) there is (b,a)
5. { (2,3) } No because there is no pair (b,a)

Antisymmetric : A relation on A is Said to be Antisymmetric if there is (a,b) then there can never ever be (b,a) unless a=b  .

it allows diagonal elements

1. {} yes because there is no pair
2. { ( 1,2) ,(3,4)} yes because there is no pair (b,a)

3, {(1,1),(2,2),(3,4)} yes because it allows diagonal elements

Asymmetric : A relation on A is said to be Asymmetric

If (a,b) belongs to R

Then (b,a) never belongs to R

Difference between Antisymmetric and Asymmetric relation is that Asymmetric does not allow diagonal elements whereas Antisymmetric Allows

1. {} yes
2. A*A no because there is (b,a) for any (a,b)
3. { (1,1),(2,3),(4,5)} no because it doesn’t allow any of the diagonal elements
4. { (2,3) ,(4,5) } yes there is no pair of (b,a )

Transitive : A relation on A said to be Transitive

If (a,b) and (b,c) belongs to R

Then (a,c) must belongs to R

1. {} there is no pair so we do not have to need check
2. R*R yes

3.{(1,2),(2,2) } yes

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