Can a relation be a partial order and an equivalence at the same time?

Question

Can a relation be a partial order AND an equivalence at the same time? For instance, if we have a set $A = \{1, 2, 3, 4, 5\}$ and a relation $R$ on $A$ defined as $R = \{(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)\}$: this relation is reflexive, anti-symmetric, symmetric, and would it be considered transitive as well? If it is considered transitive, I suppose that it is an equivalence and a partial order at the same time.

Answer 1

Two remarks:

(1) Equality is an equivalence relation which is also a partial order ($\leq$).

(2) An equivalence relation is never a strict partial order ($<$).

Answer 2

Yes, but it must be exactly like the one you propoes, that is all the equivalence classes must be of size one.

For if we had an equivalence class with at least two elements a,b we would $(a,b),(b,a)\in R$

so it would not satisfy antisymmetry