A relation '𝑅 ' is defined on ordered pairs of integers as:…

Check reflexivity: For any ordered pair (x,y), (x,y)R(x,y) would require x<x and y>y, which is impossible. So the relation is not reflexive.

Check symmetry: If (x,y)R(u,v) then x<u and y>v. For symmetry we would need u<x and v>y at the same time, which cannot hold. So the relation is not symmetric.

Check transitivity: If (x,y)R(u,v) and (u,v)R(a,b), then x<u and u<a, so x<a; and y>v and v>b, so y>b. Hence (x,y)R(a,b). The relation is transitive.

Check antisymmetry: There are no distinct pairs with both (x,y)R(u,v) and (u,v)R(x,y) because x<u and u<x cannot both hold. Thus antisymmetry holds vacuously, but antisymmetry alone does not make the relation a partial order without reflexivity.

Totality: Total order additionally requires that for any two distinct pairs either the first relates to the second or vice versa. For example, (0,0) and (1,1) are not related in either direction, so the relation is not total.

Conclusion: The relation is transitive and vacuously antisymmetric, but it is not reflexive and not symmetric. Therefore it is neither a partial order nor an equivalence relation.