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

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

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

R1 ∩ R{ (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.

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3 Comments

A relation cannot be reflexive and irreflexive simulatenously.
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But if we take R1={(1,1),(2,2)} and R2={(3,3)} then R1 intersection R2 will be phi which is irreflexive relation
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Your relation R1 and R2 is not reflexive.
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