actually, 1, 3 & 4 makes a partial order relation.
ok, so check if it is an equivalence relation:
(1) does x relate to itself? can you find and integer such that x = zx?
(2) do we have symmetry. if we can find and integer z so that y = zx, will we always be able to find an integer n so that x = ny?
(3) do we have transitivity? if we have integers n and m so that a = mb and b = nc, can we find an integer k so that a = kc?
brilliant!
think of an example.(2) (s) ,
if we multiply these we have For we have (Does this mean that we have a symmetry?
take y = 2 and x = 1. clearly y relates to x, since y = zx (where z = 2).
but does x relate to y? can we find an integer k so that x = ky, that is, 1 = k*2?
yup, that's nice. we do have transitivity(3) (t) . I multiplied again
sub.
What do you think?
no, anti-symmetric is defined by the fourth definition you have. after reading a few math books you should start to see how mathematicians define things. if anti-symmetric means "not symmetric" that is exactly how they would define it. it shouldn't be hard to come up with an example of a relation that is both not symmetric and not anti-symmetric. see here
you have to check if it is anti-symmetric. if it is, we have an order relation (since you already showed we have reflexivity and transitivity)