A relation R is defined on ordered pairs of integers as follows: (x,y) R(u,v)…

Final classification: The relation is neither a partial order nor an equivalence relation.

• Reflexive? No. For any (x,y) the condition would require x<x and y>y, which is false (for example, 0<0 is false). Thus reflexivity fails.

• Symmetric? No. If (x,y)R(u,v) means x<u and y>v, the reverse would require u<x and v>y, which cannot hold simultaneously. Example: (0,1)R(1,0) holds but (1,0)R(0,1) does not.

• Antisymmetric? Yes (vacuously). If both (x,y)R(u,v) and (u,v)R(x,y) were true, we would have x<u and u<x simultaneously, which is impossible. Since the antecedent never holds, the antisymmetry condition is satisfied vacuously.

• Transitive? Yes. If (x,y)R(u,v) (so x<u and y>v) and (u,v)R(a,b) (so u<a and v>b), then x<a and y>b, hence (x,y)R(a,b).

• Total (comparability)? No. There exist pairs that are incomparable. For example, (0,0) and (1,1) are not related in either direction, so the relation is not total.

Conclusion: Because reflexivity fails (and symmetry fails), the relation is neither a partial order (which requires reflexivity, antisymmetry, and transitivity) nor an equivalence relation (which requires reflexivity, symmetry, and transitivity).