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"If xRy and yRz": Look at the definition of R. You have x = 0 and y ≥ 0 because xRy. You have y = 0 and z ≥ 0 because of yRz. If you have xRy and yRz then you have x = 0 and z ≥ 0, therefore xRz. So R is transitive. In total you know that x = 0, y = 0 and z ≥ 0, so you also have yRx, although nobody cares much about that.


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