Q. 343.5( 4 Votes )

# For real numbers x and y, define x R y iff x – y+ √2 is an irrational number. Then the relation R is

A. reflexive

B. symmetric

C. transitive

D. none of these

Answer :

R: x R y ⟺ is irrational.

Since, x-x=0 always, therefore is irrational. Hence, (x, x)∈R ∀ x ∈ R. Hence, R is reflexive.

R is not symmetric. Proof by counter-example:

Let . Then is irrational, Therefore (x, y)∈R. But

is rational, therefore (y, x)∉R. Hence, R is not symmetric.

R is not transitive. Proof by counter-example:

Let and , then

is irrational, therefore (x, y)∈R

is irrational, therefore (y, z)∈R

is rational, therefore (x, z)∉R. Hence, R is not transitive.

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