This is a question regarding reflexivity, symmetry and transitivity.
Let $A = \{a, b, c\}$ and $R = \{(a, a), (b, b), (c, c), (c, b)\}$
From the above, I understand that A is a reflexive due to $\{(a, a), (b, b), (c, c)\}$
For symmetry, the definition states that $x,y \in R, x = y$ implies $y = x$. Hence, $(y,x) \in \ R$. Based on this, I will assume A is not symmetry as if it is to be symmetry, it needs to have $(c, b), (b, c)$ in the sets making it $R = \{(a, a), (b, b), (c, c), (c, b), (b,c)\}$. Am I right on this point?
As for transitivity, I am not sure if it applies here but from the definition, it basically means to have transitivity, I will need to have something like $(a, b), (b, c), (a, c)$. In this case, I do not see any hence it is not a transitivity.
My answer to this is that A is relexive only.
Is it also right for me to think that if the I do not see the elements in the sets that can form one of the above relations, then A will not have that relation?
Your justifications are wrong. The three relation properties are are all universally quantified predicates over elements of $A$. To show that a property does not hold, then, it suffices to find elements that satisfy the premises of the predicate but fail the conclusion.
To show that $R$ is not symmetric here, note that $(c,b)\in R$ but $(b,c)\not\in R$. However, $R$ is indeed transitive, and I leave it to you to show it by examining all possible cases.