Question

Let $A = \\{1, 2, 3, 4\\}$ and $R = \\{(1, 2), (2, 3), (1, 4)\\}$ be a relation on $A$.Let $S$ be the equivalence relation on $A$ such that $R \subseteq S$ and the number of elements in $S$ is $n$. Then, the minimum value of $n$ is _____

Answer 1

Given $A = \\{1, 2, 3, 4\\}$ and $R = \\{(1, 2), (2, 3), (1, 4)\\}$, for $R$ to be an equivalence relation, it must satisfy the following properties: reflexive, symmetric, and transitive.

Reflexivity: Add all pairs of the form $(a, a)$, where $a \in A$:
$\\{(1, 1), (2, 2), (3, 3), (4, 4)\\}$

Symmetry : Add pairs such that if $(a, b) \in R$, then $(b, a)$ must also belong to $R$:
$\\{(2, 1), (3, 2), (4, 1)\\}$

Transitivity : Ensure that if $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$. For example:
$(1, 2), (2, 3) \implies (1, 3)$
Applying this to all pairs results in:
$\\{(1, 3), (3, 1), (2, 4), (4, 2), (4, 3), (3, 4)\\}$

Combining all the above, the final relation $R$ becomes:
$R = \\{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (1, 4), (4, 1), (1, 3), (3, 1), (2, 4), (4, 2), (4, 3), (3, 4)\\}$

Thus, the total number of elements in $R$ is 16.

$\boxed{\text{Answer: } 16.}$