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 correct answer is 13.
R = {(1, 2), (2, 3), (2, 4)}
for reflexive, we need to add,
(1, 1), (2, 2), (3, 3), (4, 4)
for symmetric
if (1, 2) ∈ R then (2, 1) ∈ R
if (2, 3) ∈ R then (3, 2) ∈ R
if (2, 4) ∈ R then (4, 2) ∈ R
So set becomes
{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (2, 3), (3, 2), (2, 4), (4, 2)}
for transitive As (1, 2) ∈ R (2, 3) ∈ R
then (1, 3) ∈ R then (3, 1) ∈ R (for symmetric)
& (1, 2) ∈ R (2, 4) ∈ R
then (1, 4) ∈ R
then (4, 1) ∈ R (for symmetric)
(3, 2) ∈ R (2, 4) ∈ R
then (3, 4) ∈ R then (4, 3) ∈ R (for symmetric)
so 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)}
so 13 new elements are added
⇒ n = 13