Question

if R1 and R2 are reflexive relations on set A, then is R1 intersection R2 irreflexive?

Comments

Reflexive relations are closed under UNION, thus Reflexive UNION Reflexive is always REFLEXIVE.

Answer 1

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.

Comments

Your relation R1 and R2 is not reflexive.

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No it's not let me explain

Let

A={1,2,3}

R1={(1,1),(2,2),(3,3)}

R2={(1,2),(2,3),(3,1)}

Then R1 intersection R2 will be { } i.e. void empty relation which can never be reflexive

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Bro u didn't follow the condition only in ur example R1 a d R2 none of them are reflexive as a reflexive reltn must contain a element defined from itself to itself which isn't there