What's the difference of $xRy=1$ on $\mathbb {Z}$ or $\mathbb {R}$? Suppose we define the relation xRy iff xy=1, is this relation:
Reflexive?
Irreflexive?
Symmetric?
Antisymmtric?
Transitive?
From my understanding on Z (integers), the relation is symmetric and transitive.
Symmetric: if xy=1 then yx=1 is true
Transitive: if xy=1 then yz=1, so xz=1
Neither Reflexive or irreflexive: xx=1 iff x=1, so there is at least one instance, not all instances, where xRx, so it's neither (if I am applying this correctly - it's reflexive if xx=1 for all x, and irreflexive if xx!=1 for all x)
However, what's the difference between asking if this is on the Real (R) set or the integer set (Z)?  Are they the same?
 A: There’s a big difference between doing this for real numbers vs integers. For one thing, in the integers $xRy\Rightarrow x=y$,  which isn’t true for te real numbers. We have $2R0.5$, $0.5R2$, but not $2R2$ so for the reals it is not transitive.
You have the right answers but the wrong analysis for the integers. In addition to $1R1$ there is also $-1R-1$
A: Of course not. When you extend the set, you could never be guaranteed to keep the property of a relation. It's possible to keep the property only when the extension is somewhat natural and compatible with the property by which you defined the relation.
In this case, when you consider in $R$, the transitivity is broken, (you can easily notice that.)
A: Reflexive: Fails in both Integers and Reals.
Symmetic: Passes in both Integers and Reals.
Transitive: Passes in Integers but fails in Reals.  
We can write the relation as $R_A=\{(x,x^{-1}): x\in A\}$ where $A$ is either $\Bbb Z$ or $\Bbb R$.   As you noticed, when under the Integers, the relation is $R_{\Bbb Z}=\{(1,1),(-1,-1)\}$ as only $1$ and $-1$ have multiplicative inverse (which are themselves) and this trivially transitive.   However, in the Reals there are many more entries in $R_\Bbb R$, which preserves symmetry but breaks transitivity [because if $xy=1\wedge yz=1$ then $x=z$, but $xx=1$ only if $x=\pm 1$].
