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⊂S and the number of elements in S is n. Then, the minimum value of n is_______.

To find the minimum value of n, we need to understand what an equivalence relation is. An equivalence relation on a set A is a relation which is reflexive, symmetric, and transitive.

1. Reflexive: For all a in A, (a, a) is in S. This means every element must be related to itself. So, for A={1,2,3,4}, we need at least 4 elements in S: {(1,1), (2,2), (3,3), (4,4)}.

2. Symmetric: For all a, b in A, if (a, b) is in S, then (b, a) is in S. This means if 1 is related to 2, then 2 must be related to 1. We already have {(1,2),(2,3),(1,4)} in R, so we need to add their symmetric pairs to S: {(2,1), (3,2), (4,1)}.

3. Transitive: For all a, b, c in A, if (a, b) and (b, c) are in S, then (a, c) must be in S. This means if 1 is related to 2 and 2 is related to 3, then 1 must be related to 3. We already have {(1,2),(2,3)} in R, so we need to add (1,3) to S. Similarly, we have {(1,2),(2,3),(1,4)} in R, so we need to add (1,3), (2,4), and (3,4) to S.

So, the minimum number of elements in S is the sum of the elements needed for reflexivity, symmetry, and transitivity. This is 4 (for reflexivity) + 3 (for symmetry) + 3 (for transitivity) = 10. Therefore, the minimum value of n is 10.