Let $\;C$ = set of cities.
The relation $\,S=\big\{(x,y)\;|\;x\in C\text{ and }y\in C$ are less than $50$ miles from each other$\big\}$
to me understanding is :
- reflexive: all cities are less than $50$ miles from themselves
- symmetry: if $x$ is less than $50$ miles from $y$, then $y$ must also be less be $50$ miles from $x$, but $y$ does not need to be igual $x$
- not transitive：if $x$ is less than $50$ miles from $y$ and $y$ is less than $50$ miles from $z$, then $x$ might not be less than $50$ miles from $z$.
For example:
$x-y = 20$ miles
$y-z = 40$ miles
$x-z = 60$ miles
Partial Order is reflexive, Antisymmetric and transitive.
Antisimmetry$:\big((a,b) \in S \big)\land \big((b,a) \in S \big) \Rightarrow a=b $
which in this case seems not to be true because of symmetric property you mentioned.
Note that I don't know what set you are working with, but an example where symmetry need not hold would be directed graphs, because path from a to b is not necessarily equal to path from b to a.
but it's obviously not transitive so no, it's not a partial order.