Write the smallest equivalence relation on the set `A={1,\ 2,\ 3}` .

Answer

`R={(1,1),(2,2),(3,3)}`

Explanation

The smallest equivalence relation on the set` {A}={1,2,3}` is,
` R={(1,1),(2,2),(3,3)}`.


it is reflexive as ` forall x in A,(x, x) in R.`


relation R is symmetric as `forall (x, y) in R`
`Rightarrow EE (y, x) in R ; forall x, y in A.`


R is transitive as `forall (x, y) in R,and (y, z) in R`.
`Rightarrow EE(x, z) in R ; forall x, y, z in A`.


Hence our relation is an equivalance relation.