Given the relation R = {(1, 2), (2, 3)} on the set A = {1, 2, 3}, add a minimum number ordered pairs so that the enlarged relation is symmetric, transitive and reflexive.
Given is:
R = {(1, 2), (2, 3)} on the set A.
A = {1, 2, 3}
Right now, we have
R = {(1, 2), (2, 3)}
Symmetric Relation:
We know (1, 2) ∈ R
Then, (2, 1) ∈ R
Also, (2, 3) ∈ R
Then, (3, 2) ∈ R
So, add (2, 1) and (3, 2) in R, so that we get
R’ = {(1, 2), (2, 1), (2, 3), (3, 2)}
Transitive Relation:
We need to make the relation R’ transitive.
So, we know (1, 2) ∈ R and (2, 1) ∈ R
Then, (1, 1) ∈ R
Also, (2, 3) ∈ R and (3, 2)
Then, (2, 2) ∈ R
Also, (2, 1) ∈ R and (1, 2) ∈ R
Then, (2, 2) ∈ R
Also, (3, 2) ∈ R and (2, 3) ∈ R
Then, (3, 3) ∈ R
Add (1, 1), (2, 2) and (3, 3) in R’, we get
R’’ = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}
Thus, we have got a relation which is reflexive, symmetric and transitive.
R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1), (2, 3), (3, 2)}
The ordered pair added are (1, 1), (2, 2), (3, 3), (3, 2).