Question

A = {1, 2, 3, 4} , R = {(1, 2), (2, 3), (2, 4)} R ⊆ S and S is an equivalence relation then the minimum number of elements to be added to R is n, then the value of n is?

Answer 1

The total number of elements is 13. The initial relation is R = {(1, 2), (2, 3), (2, 4)}. To make R reflexive, the elements (1, 1), (2, 2), (3, 3), and (4, 4) must be added. To make R symmetric, for every (a, b) in R, (b, a) must also be in R. This adds (2, 1), (3, 2), and (4, 2). The set thus becomes {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2)}. For transitivity, if (a, b) ∈ R and (b, c) ∈ R, then (a, c) must be in R. Considering the current set: (1, 2) and (2, 3) imply (1, 3). By symmetry, (3, 1) is added. (1, 2) and (2, 4) imply (1, 4). By symmetry, (4, 1) is added. (3, 2) and (2, 4) imply (3, 4). By symmetry, (4, 3) is added. The final set S = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2), (1, 3), (3, 1), (1, 4), (4, 1), (3, 4), (4, 3)}. This results in 13 new elements being added, so n = 13.