Given the relation $R={(1,2),(2,3)}$ on the set $A={1,2,3}$ the minimum number of ordered pairs required to make R an equivalence relation is

Question

If we add (1,1), (2,2), (3,3), we get a reflexive relation

If we add (2,1), (3,2) we get a symmetric relation

If we add (1,3), we get a transitive relation.

All together, we have added 6 ordered pairs, so the answer should be 6. However, the answer is given as 7, so which pair am I missing?

Answer 1

Once you add $(1,3)$, you also need to add $(3,1)$ for symmetry.

You can also see that $1,2,3$ need to all be in the same equivalence class, so all ordered pairs $(x,y)$ with $x,y\in\{1,2,3\}$ need to be included. There are nine such ordered pairs.