Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.
We have the relation R such that
R = {(1, 2), (1, 1), (2, 3)}
R is defined on set A.
A = {1, 2, 3}
Recall that,
A relation R defined on a set A is called transitive if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R, ∀ a, b, c ∈ A.
For transitive relation:
Note in R,
(1, 2) ∈ R and (2, 3) ∈ R
Then, (1, 3) ∈ R
So, add (1, 3) in R.
R = {(1, 2), (1, 1), (2, 3), (1, 3)}
Now, we can see that R is transitive.
Hence, the ordered pair to be added is (1, 3).