Given the set ${1, 2, 3, 4}$, and that $R$ is an equivalence relation. Since ${((1, 2), (1, 3))} \subset R$, $R$ must contain the pairs $(1, 2)$ and $(1, 3)$. An equivalence relation must be reflexive, symmetric, and transitive.

Reflexive property: $(1, 1)$, $(2, 2)$, $(3, 3)$, $(4, 4) \in R$.

Symmetric property: Since $(1, 2) \in R$, $(2, 1) \in R$. Since $(1, 3) \in R$, $(3, 1) \in R$.

Transitive property: Since $(1, 2) \in R$ and $(1, 3) \in R$, it follows that $(2, 3) \in R$. By symmetric property, $(3, 2) \in R$.

Therefore, the elements in $R$ are: $(1, 1)$, $(2, 2)$, $(3, 3)$, $(4, 4)$, $(1, 2)$, $(2, 1)$, $(1, 3)$, $(3, 1)$, $(2, 3)$, $(3, 2)$.

Thus, the number of elements in $R$ is $10$.