David G. answered • 09/07/19
Patient, Effective math/statistics tutor
A partial order R on a set is a relation which is reflexive, transitive, and antisymmetric.
An example is ≥.
A total order is a partial order R with the property that for every pair of element x and y is it true that
either x R y or y R x. The partial order ≥ is also a total order, because either x ≥ y or y ≥ x.
However, not every total order is a partial order.
For example, on the set of integers, the relation "divides evenly", usually indicated by |, is a partial order. To verify this, we check that every number divides itself evenly (reflexive); if x divides y, and y divides z, then x divides z (as 3 divides 6, and 6 divides 18, so 3 divides 18) (transitive); and if x divides y and y divides x, this implies that x = y (antisymmetric).
However, "divides evenly" is not a total order. For example, 3 does not divide 5, and 5 does not divide 3.
