Final answer:
The relation 'R' defined as (x,y)R(u,v) if xv is neither a partial order nor an equivalence relation because it does not satisfy all the necessary conditions such as reflexivity and symmetry.
Step-by-step explanation:
The relation 'R' on ordered pairs of integers is defined as (x,y)R(u,v) if xv. To determine whether this is a partial order or an equivalence relation, we need to examine its properties.
A relation is a partial order if it is reflexive, antisymmetric, and transitive. It is an equivalence relation if it is reflexive, symmetric, and transitive. Based on the definition of R:
- Reflexivity is not satisfied because for an ordered pair (x,x), it is stated that x×x ≠ x×x, which is false.
- Antisymmetry could be satisfied because if (x,y)R(u,v) and (u,v)R(x,y) then this implies x×v = y×u and u×y = v×x which can lead to the conclusion that x = u and y = v under certain conditions.
- Transitivity is satisfied because if (x,y)R(u,v) and (u,v)R(s,t) then x×v = y×u and u×t = v×s which implies x×t = y×s.
- Symmetry is not satisfied because if (x,y)R(u,v) implying x×v = y×u it does not mean that (u,v)R(x,y).
However, since not all of the required conditions for a partial order or an equivalence relation are met, the relation 'R' is neither a partial order nor an equivalence relation.
Thus, the correct answer is a. Neither a partial order nor an equivalence relation.