If $$R$$ is the smallest equivalence relation on the set $$\{1, 2, 3, 4\}$$ such that $$\{(1, 2), (1, 3)\} \subset R$$, then the number of elements in $$R$$ is ______.

We need to find the number of elements in the smallest equivalence relation $$R$$ on $$\{1, 2, 3, 4\}$$ such that $$\{(1,2), (1,3)\} \subset R$$.

An equivalence relation must be reflexive, symmetric, and transitive.

Given: $$(1,2)$$ and $$(1,3)$$ must be in $$R$$.

Reflexive property requires: $$(1,1), (2,2), (3,3), (4,4)$$ must all be in $$R$$.

Since $$(1,2) \in R$$, we need $$(2,1) \in R$$.

Since $$(1,3) \in R$$, we need $$(3,1) \in R$$.

Since $$(2,1) \in R$$ and $$(1,3) \in R$$, we need $$(2,3) \in R$$.

By symmetry of $$(2,3)$$: $$(3,2) \in R$$.

Check: $$(3,1) \in R$$ and $$(1,2) \in R$$ gives $$(3,2) \in R$$ — already included.

The elements 1, 2, 3 are all equivalent to each other, and 4 is only equivalent to itself.

$$R = \{(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (1,3), (3,1), (2,3), (3,2)\}$$

$$|R| = 10$$

The correct answer is Option 1 — $$10$$.