# For real numbers x and y, define x R y iff x – y+ √2 is an irrational number. Then the relation R isA. reflexiveB. symmetricC. transitiveD. none of these

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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