Symmetric Relations and Cycles

Question

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.

Answer 1

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.

Answer 2

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 $(a,b)\leftrightarrow(b,a)$.