Antisymmetric Relation 
Determine whether the relation R on the set of all people is antisymmetric.
      (a) a is taller than b.
      (b) a and b are born on the same day.
      (c) a has the same first name as b.
      (d) a and b have a common grandparent.

My understanding of antisymmetric relations: A relation R defined on a set S and having the property that whenever x R y and y R x then x = y.  (Is this correct? Every explanation of antisymmetry I find hard to understand.)
That being said, with (A)  It would seem to me if (a) is taller than (b), (b) can't be taller than (a), which would not satisfy antisymmetry since (a), (b) have to be related and (b), (a) also need to be related but it is impossible for (a) to be both taller and shorter than (b).  So help me understand - why would the first question be antisymmetric?
 A: Your statement of the definition of antisymmetry is correct, but your understanding of it is not. A relation $R$ on a set $A$ is antisymmetric if it has the following property: for any $a,b\in A$, if $a\,R\,b$ and $b\,R\,a$, then $a=b$. That if is very important. If you find an $a$ and a $b$ such that $a\,R\,b$ and $b\,\not R\,a$, the definition says nothing about that $a$ and $b$. Similarly, it says nothing about $a$ and $b$ if $a\,\not R\,b$ and $a\,R\,b$, or if $a\,\not R\,b$ and $b\,\not R\,a$. Antireflexivity of $R$ just says that $A$ does not contain two distinct elements $a$ and $b$ such that $a\,R\,b$ and $b\,R\,a$.
Thus, the relation in (a) is antisymmetric: there aren’t two different people, each of whom is taller than the other. If $a$ is taller than $b$ and $b$ is taller than $a$, then $a=b$ is a vacuously true statement. To say the same thing in a slightly different way, for this relation no violation of antisymmetry is possible. A violation would be an $a$ and $b$ such that $a$ was taller than $b$ and $b$ was taller than $a$, but $a$ and $b$ were different people. It’s not possible to have an $a$ and $b$ such that $a$ is taller than $b$ and $b$ is taller than $a$ in the first place, never mind whether or not $a$ and $b$ are the same person.
