why is "x − y is a rational number" is not antisymmetric while "x = 2y" is antisymmetric

the proof I found for the first one is that "Since (1, −1) and (−1, 1) are both in R,

the relation is not antisymmetric." While the proof for the other was, "To see that it is antisymmetric, suppose that x = 2y, and y = 2x. Then y = 4y, from

which it follows that y = 0 and hence x = 0. Thus the only time that (x, y) and (y, x) are both in R

is when x = y (and both are 0)".

now the question is, why is it enough for the first equation to give an example that lies outside the relation to proving that it is **NOT** antisymmetric. Whereas for the second equation it was enough to give one example to prove that it is antisymmetric, even though in the relation R there exists element which is not antisymmetric

why is "x − y is a rational number" is not antisymmetric while "x = 2y" is antisymmetric

the proof I found for the first one is that "Since (1, −1) and (−1, 1) are both in R,

the relation is not antisymmetric." While the proof for the other was, "To see that it is antisymmetric, suppose that x = 2y, and y = 2x. Then y = 4y, from

which it follows that y = 0 and hence x = 0. Thus the only time that (x, y) and (y, x) are both in R

is when x = y (and both are 0)".

now the question is, why is it enough for the first equation to give an example that lies outside the relation to proving that it is **NOT** antisymmetric. Whereas for the second equation it was enough to give one example to prove that it is antisymmetric, even though in the relation R there exists element which is not antisymmetric