Given, for real numbers x and y,

xRy => x - y + √2 is an irrational number.

Let R is a binary relation on real numbers x and y.

Now, R is transitive iff for all (x, y) ∈ R and (y, z) ∈ R implies (x, z) ∈ R

Given, xRy => x - y + √2 is irrational    ............1

and yRz => y - z + √2 is irrational       ............2

Add equation 1 and 2, we get

   (x - y + √2) + (y - z + √2) is irrational

= x - z + √2 is irrational

= xRz is irrational

So, the relation R is transitive.