Symmetric Relations and Cycles Let's say I have the set:
R = {(a,b),(b,c),(c,d),(d,a)}
If you visualize this in graph form, it forms a cycle.
My question is, is this already a symmetric relation, or do I have to add (b,a),(b,c),(d,c),(d,a) to make it symmetric. 

By definition, if the relation is "symmetric" and contains $(a,b)$ it must also contain $(b,a)$.
There's nothing particularly deep about that; it is just how the word "symmetric" happens to be used about relations.
As you (almost) point out, the relation has other symmetries than the one represented by looking at it backwards -- for example the relation is preserved by certain permutations of its underlying set. This would usually be called a "symmetry" of it, except that there's a strong convention that when we speak about relations, "symmetry" is reserved for speaking about the correspondence $(a,b)\leftrightarrow(b,a)$.
