Reflexive Relation on Set
Reflexive Relation on Set : A relation R on set A is said to be reflexive relation if every element of A is related to itself.R is a reflexive $\Leftrightarrow $ (a,a) $\in $ R for all a $\in $ A.
An example of a reflexive relation is the relation "is equal to" on the set of real numbers, since every real number is equal to itself.
For example, let set A = {1,2,3} then R= {(1,1),(2,2),(3,3),(1,3),(2,1)} is a reflexive relation on set A. But R1= {(1,1),(3,3),(2,1),(3,2)} is not a reflexive relation on A because 2 $\in $ A but (2,2) $\notin $ R1.
Examples on Reflexive Relation on Set
Example 1 : R is a relation in A = {set of straight lines in a plane}A = { l1,l2,l3,.....ln} which is defined as R= {(l1,l1),(l2,l2),....(ln,ln)} Since each line in a plane is parallel to itself, so this type of relation is a reflexive relation.
Example 2 : Why is R = {(1,1), (2,2), (3,3)} not reflexive on {1,2,3,4}?
Solution : As we know that R is a reflexive $\Leftrightarrow $ (a,a) $\in $ R for all a $\in $ A.
But in the above relation R, (4,4) ordered pair is missing so R = {(1,1), (2,2), (3,3)} not reflexive on {1,2,3,4}.
Example 3 : Let A = {1, 2, 3, 4,5,6,7,8,9,10} and define R = {(a, b) | a divides b} We saw that R was reflexive since every number divides itself Let A = {1, 2, 3, 4,5,6,7,8,9,10} and define R ={(1,1),(2,2),(2,3),(3,2),(4,4)} We saw that R is not reflexive since the elements 5,6,7 and 8 are missing .
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