Let $\mathbf{X}$ be the set of all residents in New Jersey. Determine which are equivalance relations:
a) $x\sim y$ provided $y$ has the same natural parents as $x$
b) $x\sim y$ provided $y$ lives within 5 miles of $x$
c) $x\sim y$ provided $y$ has same birth date as $x$
d) $x\sim y$ provided $y$ is the brother of $x$
I know i have to either prove all reflective, symmetric and transitive properties or disprove one. I'm struggling how to show my reasoning for it.
For example for (b) I can see it fails the transitive prop as say $z$ could be 5 miles east of $y$ which is 5 miles east of $x$ thus $x$ isn't within 5 miles of $x$ and (d) is not symmetric as $x$ could be the sister of $y$.
Thanks