Let R = {(1, 2), (2, 3)} be a relation defined on set {1, 2, 3}. The minimum number of ordered pairs required to be added in R, such that enlarged relation becomes an equivalence relation is

• 3
• 5
• 7
• 9

To determine the minimum number of ordered pairs needed to make the relation R = {(1, 2), (2, 3)} an equivalence relation on the set {1, 2, 3}, we first need to understand the properties of an equivalence relation. An equivalence relation must be reflexive, symmetric, and transitive.

Current Status of R

The existing pairs in R are:

• (1, 2)
• (2, 3)

Reflexivity

For R to be reflexive, we need to include the pairs (1, 1), (2, 2), and (3, 3). This adds 3 pairs.

Symmetry

Next, we check for symmetry. Since we have (1, 2), we need to add (2, 1). For (2, 3), we need (3, 2). This adds 2 more pairs.

Transitivity

Finally, we need to ensure transitivity. With the pairs we have, we need to add (1, 3) and (3, 1) to connect all elements. This adds 2 more pairs.

Total Pairs Needed

Now, let's sum up the pairs:

• 3 pairs for reflexivity
• 2 pairs for symmetry
• 2 pairs for transitivity

This results in a total of 7 pairs needed to make R an equivalence relation.

Thus, the minimum number of ordered pairs required to be added to R is 7.