relations reflexive, transitive, symmetric

Question

Determine whether the relation R on the set of real numbers is reflexive, symmetric, and transitive,

where

xRy ←→ x − y = 0

Answer 1

This relation is reflexive, symmetric and transitive.

To show R is reflexive, we need to show that xRx. For any real number x, it is always true that x-x=0, so xRx, and we have reflexivity.

To show R is symmetric, we must show that if xRy, then yRx. If xRy, then x-y=0. This also gives us that x=y and hence 0 = y-x, meaning yRx.

To show R is transitive, we must show that if xRy and yRz, then xRz. If xRy and yRz, then x-y=0 and y-z=0. This means that x=y and that y=z, and hence x=z. Thus x-z = x-(x), which we know is 0, so x-z=0, and xRz.