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
