Partial Order Relations

Question

Let A = Z. Let R be defined on A × A where xRy if x ≤ y . Prove that R is a partial order relation.

Answer 1

To be a partial order, a relation needs to be (a) homogeneous, (b) transitive, and (c) antisymmetric.

x ≤ y is homgeneous, because it's binary relation on a set X and itself. It's transitive because if x ≤ y and y ≤ z, then x ≤ z. It's antisymmetric because if x ≤ y, and y ≤ x, then x = y (the difference between antisymmetric and asymmetric is that the former allows identify whereas the latter is stricter: if xRy, then ~yRx).

Hope that helps!