Question

 Give a relation which is both a partially ordered relation and an equivalence relation on a set.

Answer 1

The only relations that are both symmetric and anti-symmetric are identity relations of the form {(a,a),(b,b),…}. Hence {(1,1),(2,2),(3,3),(4,4)} is the only relation on the set {1,2,3,4} that is reflexive, symmetric, and anti-symmetric. Clearly it is also transitive, and hence it is the only relation that is both a partial order and an equivalence relation on the set {1,2,3,4}

The same argument goes for any set S: The only relation that is both a partial order and an equivalence relation is the identity relation R={(x,x)∣x∈S}.