That is correct. An equivalence relation must be reflexive, symmetric and transitive.

Your relation, however, is R. That means R must be reflexive, symmetric and transitive because R represents a relation on {a, b, c, d, e, f}. You do not imply assign the individual pairs as being symmetric etc. It must be the entire relation.

R= {(a,a),(a,b),(a,c),(b,a),(b,b),(b,c),(c,a),(c,b),( c,c),(d,d),(d,e),(e,d),(e,e),(f,f)}

A relation is reflexive if it contains all pairs of the form (a,a). Therefore, if R represents a relation on {a,b,c,d,e,f} it must contain (a,a), (b,b), (c,c), (d,d), (e,e) and (f,f), which your relation does.

To test for symmetry, you have to make sure no pair in the relation without the opposite. Meaning is (a,b) is in the relation, you must also have (b,a). Your relation is symmetric.

A relation is transitive if if whenever the relation contains (a,b) and (b,c), it also contains (a,c). This also menas that if it contains the pairs (a,b) and (b,a), it must also have (a,a) and (b,b). Your relation R is therefore also transitive and a equivalence relation.

I hope this helps!