Questions on set relations

I have some question i hope anyone could clarify for me.
I am asked whether a relation R on set of all real numbers is
reflixve, symmetric, antisymetric and/or transitive where (x,y) E R
iff:

I have the formal defintions for each type I'm still not understanding
them.

one the question is
x=+-y, I reason that this is reflexive because this relation is same
as (x+y=0 or x-y=0) and if
I use x=1 as example, then 1+1!=0 but 1-1=0 so its true.

I say this is not symmetric because: if I use (1,2) by definition
(2,1) would have to be in it, but
1+2 !=0 or 1-2!=0 also. So that case is false. Remember that this on
the set of all real numbers.
So is my reason okay?

If you could explain also why this is antisymetric, and transitive
also that would really put off on a good start. I have a bunch of
these and to be stuck on these first few isn't making feel too great.

Thanks much for you time for clarification.