GATE CS 2006 Q4 — Sets, Relations and Functions

To determine the nature of the relation $R$ defined on $\mathbb{Z} \times \mathbb{Z}$, let us analyze its properties.

The relation is defined as:
$$(x,y)R(u,v) \iff x < u \text{ and } y > v$$

1. Reflexivity

For $R$ to be reflexive, every element $(x, y)$ must be related to itself, i.e., $(x, y) R (x, y)$ must hold for all $x, y \in \mathbb{Z}$.

Applying the definition:
$$(x, y) R (x, y) \iff x < x \text{ and } y > y$$

Since $x < x$ is always false for any integer $x$, the relation $R$ is not reflexive (in fact, it is irreflexive).

Conclusion on Reflexivity:

• A Partial Order must be reflexive, antisymmetric, and transitive.
• An Equivalence Relation must be reflexive, symmetric, and transitive.

Since $R$ is not reflexive, it can be neither a partial order nor an equivalence relation.

2. Transitivity (Optional verification)

If $(x, y) R (u, v)$ and $(u, v) R (w, z)$, then:
$$x < u \text{ and } y > v$$
$$u < w \text{ and } v > z$$

By the transitive property of inequality on integers:
$$x < u < w \implies x < w$$
$$y > v > z \implies y > z$$

Thus, $(x, y) R (w, z)$, which means $R$ is transitive. However, the lack of reflexivity prevents it from being a partial order or an equivalence relation.

Therefore, the correct option is A.