To make $R$ symmetric, we need to add $(b, a)$ and $(c, b)$ to $R$. So, $R = \{(a, b), (b, c), (b, a), (c, b)\}$.
To make $R$ transitive:
Since $(a, b) \in R$ and $(b, c) \in R$, we need to add $(a, c)$ to $R$.
Now, $R = \{(a, b), (b, c), (b, a), (c, b), (a, c)\}$.
Since $(b, a) \in R$ and $(a, c) \in R$, we need to add $(b, c)$ to $R$. But $(b, c)$ is already in $R$.
Since $(c, b) \in R$ and $(b, a) \in R$, we need to add $(c, a)$ to $R$.
Now, $R = \{(a, b), (b, c), (b, a), (c, b), (a, c), (c, a)\}$.
Since $(a, b) \in R$ and $(b, a) \in R$, we need to add $(a, a)$ to $R$.
Since $(b, c) \in R$ and $(c, b) \in R$, we need to add $(b, b)$ to $R$.
Since $(c, a) \in R$ and $(a, c) \in R$, we need to add $(c, c)$ to $R$.
So, $R = \{(a, b), (b, c), (b, a), (c, b), (a, c), (c, a), (a, a), (b, b), (c, c)\}$.
The elements that must be added are $(b, a), (c, b), (a, c), (c, a), (a, a), (b, b), (c, c)$.
Therefore, the minimum number of elements that must be added is $7$.