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. 

Symmetric requires that $(x,y)\in R\implies (y,x)\in R$, your relation does not have that. Remember that the different relation properties such as reflexive, transitive, et cetera are not implied by one another, they are all independent.
In order that your relation be symmetric you need to check that it satisfies the definition as noted above. Since you lack, for example, $(b,a)$, it fails to satisfy the definition of a symmetric relation.
