# 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).

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