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Question:

For real numbers x and y, we define xRy if x-y+√5 is an irrational number. The relation R is Reflexive, Transitive, Symmetric, None of these?

Reflexive

Symmetric

Transitive

None of these

Solution:

x∈R ⇒ x-x+√5=√5 is an irrational number. ∴ (x,x)∈R Hence, R is reflexive (√5,1)∈R because √5-1+√5=2√5-1 which is an irrational number. ∴(1,√5)∈R Hence, R is not a symmetric We have, (√5,1)(1,2√5)∈R because √5-1+√5=2√5-1, if 1-2√5+√5=1-√5 are irrational number. ∴(√5,2√5)/∈R Hence, R is not transitive.