symmetric/antisymmetric according to both the text and my professor, these properties are not mutually exclusive.
i.e. a relation can be both symmetric and antisymmetric.
I understand the properties themselves, but I don't understand the reason that one would adopt this naming convention if a relation can have both of these properties.  assuming that the labeling isn't a deliberate attempt to confuse, there must be some deeper reason I don't grasp.  can anyone explain what that would be?  
 A: In topology a set can be open, closed or neither open nor closed. So “closed” is not the negation of “open”, which does not agree with common language.
In mathematics what's important are the definitions; the word “antisymmetric” does not denote the negation of “symmetric”, for which “not symmetric” suffices. Perhaps it's not the best terminology, but by now it's standard.
The concept expressed by “antisymmetric” is that you can draw conclusions from the fact that the pairs $(a,b)$ and $(b,a)$ both belong to the relation, precisely that if this happens, then $a=b$.
Note that “not symmetric” is expressed by existential quantifiers: there exist $a$ and $b$ such that $(a,b)$ belongs to the relation and $(b,a)$ does not belong to the relation. On the contrary, “antisymmetric” is expressed with universal quantifiers: for all $a,b$ in the set, if $(a,b)$ and $(b,a)$ belong to the relation, then $a=b$.
So a relation can be both symmetric and antisymmetric. A widely used relation enjoys both properties (see the spoiler below).

 Equality is symmetric and antisymmetric

