Question

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

Answer 1

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