Reflexive Relation

A relation ‘R’ on a set ‘A’ is said to be reflexive if (x,x)∈R ∀x∈A

Example #

A={1,2,3} R must have (1,1),(2,2),(3,3) in order to be reflexive.

R={(x,y) ; x-y is an integer} Let’s check if R is a reflexive relation on the set of natural numbers. xRx is true ∀x∈N because x-x=0 is an integer. Therefore ‘R’ is reflexive on the set of N.

How many reflexive relations possible? #

If |A|=n Number of reflexive relations: 2^(n²-n)

Example #

A={1,2} |A|=2 A×A=

List of all reflexive relations on A:

1. {(1,1),(2,2)}
2. {(1,1),(2,2),(1,2)}
3. {(1,1),(2,2),(2,1)}
4. {(1,1),(2,2),(1,2),(2,1)}

2^(2²-2) = 4 possible relations