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

(x, y) R (u, v) if x < u and y > v. Then R is

The correct answer is Neither a partial order nor an equivalence relation

Key Points

Let's reevaluate the properties of the relation R :(x,y)R(u,v) if <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;"><</span><span style="font-family:courier new,courier,monospace;">�</span> x<v and <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">></span><span style="font-family:courier new,courier,monospace;">�</span> y>v.

• Reflexivity: For any ordered pair (�,�) (x,y), it is not possible for both �<� x<y and <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">></span><span style="font-family:courier new,courier,monospace;">�</span> y>y to be true simultaneously. Therefore, � is not reflexive.
• Antisymmetry: If <span style="font-family:courier new,courier,monospace;">(</span><span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">,</span><span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">)</span><span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">(</span><span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">,</span><span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">)</span> (x,y)R(u,v) and (�,�)�(�,�) , then �<� x<v and <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">></span><span style="font-family:courier new,courier,monospace;">�</span> y>v imply �<� u<v and <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">></span><span style="font-family:courier new,courier,monospace;">�</span> v>y. However, this does not necessarily mean that (�,�)=(�,�) (x,y)=(u,v). Therefore, � is not antisymmetric.
• Transitivity: If (�,�)�(�,�) (x,y)R(u,v) and (�,�)�(�,�) (u,v)R(w,z), then �<� x<v, <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">></span><span style="font-family:courier new,courier,monospace;">�</span> y>v, <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;"><</span><span style="font-family:courier new,courier,monospace;">�</span> u<z, and �>� v>z. Combining these, we can deduce �<� x<z and <span style="font-family:courier new,courier,monospace;">�</span><span style="font-family:courier new,courier,monospace;">></span><span style="font-family:courier new,courier,monospace;">�</span> y>z. Therefore, � is transitive.
• A binary relation is an equivalence relation on a nonempty set S if and only if the relation is reflexive(R), symmetric(S) and transitive(T).
• A binary relation is a partial order if and only if the relation is reflexive(R), antisymmetric(A) and transitive(T).
• From the given relation, it is neither partial order nor equivalence relation.

So, the correct answer is indeed: 1) Neither a partial order nor an equivalence relation.