Let's Take a Example
A={1,2,3}
A $\times$ A ={ (1,1)(2,2)(3,3)(1,2)(2,1)(1,3)(3,1)(2,3)(3,2) }
Reflexive Relation :- A relation R on a set A is said to be Reflexive if (xRx)∀x∈A
$\underbrace{(1,1)(2,2)(3,3) }_{n}$$\underbrace{(1,2)(2,1)(1,3)(3,1)(2,3)(3,2) }_{n^{2}-n}$
All diagonal elements (1,1)(2,2)(3,3) should be present in every Reflexive relation.
Now Take any two relation on set A
R1={ (1,1) , (2,2) , (3,3) , (1,2) }
R2 ={ (1,1) , (2,2) , (3,3) , (2,1) }
R1 ∩ R2 = { (1,1) , (2,2) , (3,3) } which is Reflexive Relation.
Intersection of two reflexive relation can not be irreflexive.
Hence,Given statement " if R1 and R2 are reflexive relations on set A, then is R1 intersection R2 irreflexive? " is false.