I have read that: "nearer than 10" fails transitivity. In addition to that doesn't it fail symmetry (the example on integers)?
Thanks in advance.
suppose we replace "nearer than 10" with something a bit less ambiguous (we will assume our relation is on integers, just to be specific about which set we are relating):
a~b if (and only if) |a-b| < 10.
it is clear that this is reflexive:
a~a because |a-a| = |0| = 0 < 10.
it IS symmetric because if a~b, then |a-b|= |b-a| < 10, so b~a.
but what about transitivity: is it true that if:
a~b and b~c, that a~c?
no.
suppose a = 4, b= 13, c = 17.
then a~b because |a-b| = |4-13| = |-9| = 9 < 10.
and b~c because |b-c| = |13-17| = |-4| = 4 < 10.
however, we do NOT have a~c, because |a-c| = |4-17| = |-13| = 13 > 10.